Discrete Mathematics and Applications COT3100
|
|
|
- Linda Wilkerson
- 6 years ago
- Views:
Transcription
1 Discrete Mathematics and Applications CO3100 Dr. Ungor Sources: Slides are based on Dr. G. Bebis material. uesday, January 7, 2014
2 oundations of Logic: Overview Propositional logic: (Sections ) Basic definitions. Equivalence rules & derivations. Predicate logic (Section 1.4) Predicates. Quantified predicate expressions. Equivalences & derivations. CO Dr. Ungor 2 uesday, January 7, 2014
3 Propositional Logic Propositional Logic is the logic of compound statements built from simpler statements using Boolean connectives. Applications: Design of digital electronic circuits. Expressing conditions in computer programs. Queries to databases & search engines. CO Dr. Ungor 3 uesday, January 7, 2014
4 Propositional Logic Propositional Logic is the logic of compound statements built from simpler statements using Boolean connectives. What are Propositions? What are Boolean connectors? CO Dr. Ungor uesday, January 7, 2014
5 Definition of a Proposition CO Dr. Ungor 5 uesday, January 7, 2014
6 Definition of a Proposition A proposition (p, q, r, ) is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value thatʼs either true () or false () (never both, neither, or somewhere in between). CO Dr. Ungor 5 uesday, January 7, 2014
7 Definition of a Proposition A proposition (p, q, r, ) is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value thatʼs either true () or false () (never both, neither, or somewhere in between). In probability theory, we assign degrees of certainty to propositions. or now: rue/alse only! CO Dr. Ungor 5 uesday, January 7, 2014
8 Examples of Propositions CO Dr. Ungor 6 uesday, January 7, 2014
9 Examples of Propositions It is raining. (Given a situation.) CO Dr. Ungor 6 uesday, January 7, 2014
10 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China. CO Dr. Ungor 6 uesday, January 7, 2014
11 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 CO Dr. Ungor 6 uesday, January 7, 2014
12 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 he following are NO propositions: CO Dr. Ungor 6 uesday, January 7, 2014
13 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 he following are NO propositions: Whoʼs there? (interrogative, question) CO Dr. Ungor 6 uesday, January 7, 2014
14 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 he following are NO propositions: Whoʼs there? (interrogative, question) La la la la la. (meaningless interjection) CO Dr. Ungor 6 uesday, January 7, 2014
15 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 he following are NO propositions: Whoʼs there? (interrogative, question) La la la la la. (meaningless interjection) Just do it! (imperative, command) CO Dr. Ungor 6 uesday, January 7, 2014
16 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 he following are NO propositions: Whoʼs there? (interrogative, question) La la la la la. (meaningless interjection) Just do it! (imperative, command) Yeah, I sorta dunno, whatever... (vague) CO Dr. Ungor 6 uesday, January 7, 2014
17 Examples of Propositions It is raining. (Given a situation.) Beijing is the capital of China = 5 he following are NO propositions: uesday, January 7, 2014 Whoʼs there? (interrogative, question) La la la la la. (meaningless interjection) Just do it! (imperative, command) Yeah, I sorta dunno, whatever... (vague) CO Dr. Ungor (expression with a non-true/false value) 6
18 Boolean operators Negation (NO) Implication (I) Conjunction (AND) Disjunction (OR) Exclusive-Or (XOR) Bi-conditional (I) CO Dr. Ungor uesday, January 7, 2014
19 Operators / Connectives An operator or connective combines one or more operand expressions into a larger expression. (e.g., + in numeric exprs.) Unary operators take 1 operand (e.g., -3); binary operators take 2 operands (e.g., 3 4). Propositional or Boolean operators operate on propositions or truth values instead of on numbers. CO Dr. Ungor 8 uesday, January 7, 2014
20 he Negation Operator he unary negation operator (NO) transforms a prop. into its logical negation. Example: If p = I have brown hair., then p = I do not have brown hair. ruth table for NO: CO Dr. Ungor 9 uesday, January 7, 2014
21 he Conjunction Operator he binary conjunction operator (AND) combines two propositions to form their logical conjunction. E.g., If p= I will have salad for lunch. and q= I will have steak for dinner., then p q= I will have salad for lunch and I will have steak for dinner. CO Dr. Ungor 10 uesday, January 7, 2014
22 Conjunction ruth able CO Dr. Ungor 11 uesday, January 7, 2014
23 Conjunction ruth Note that a conjunction p 1 p 2 p n of n propositions will have 2 n rows in its truth table. able and operations together are universal, i.e., sufficient to express any truth table! CO Dr. Ungor 11 uesday, January 7, 2014
24 he Disjunction Operator he binary disjunction operator (OR) combines two propositions to form their logical disjunction. p= hat car has a bad engine. q= hat car has a bad carburetor. p q= Either that car has a bad engine, or that car has a bad carburetor. CO Dr. Ungor 12 uesday, January 7, 2014
25 Disjunction ruth able CO Dr. Ungor 13 uesday, January 7, 2014
26 Disjunction ruth Note that p q means that p is true, or q is true, or both are true! So this operation is also called inclusive or, because it includes the able possibility that both p and q are true. and together are also universal. CO Dr. Ungor 13 uesday, January 7, 2014
27 A Simple Exercise Let p= It rained last night, q= he sprinklers came on last night, r= he lawn was wet this morning. ranslate each of the following into English: p = r p = r p q = CO Dr. Ungor 14 uesday, January 7, 2014
28 A Simple Exercise Let p= It rained last night, q= he sprinklers came on last night, r= he lawn was wet this morning. ranslate each of the following into English: p = It didn t rain last night. r p = r p q = CO Dr. Ungor 14 uesday, January 7, 2014
29 A Simple Exercise Let p= It rained last night, q= he sprinklers came on last night, r= he lawn was wet this morning. ranslate each of the following into English: p = r p = r p q = CO Dr. Ungor 14 uesday, January 7, 2014
30 A Simple Exercise Let p= It rained last night, q= he sprinklers came on last night, r= he lawn was wet this morning. ranslate each of the following into English: p = r p = he lawn was wet this morning, and it didn t rain last night. r p q = CO Dr. Ungor 14 uesday, January 7, 2014
31 A Simple Exercise Let p= It rained last night, q= he sprinklers came on last night, r= he lawn was wet this morning. ranslate each of the following into English: p = r p = r p q = CO Dr. Ungor 14 uesday, January 7, 2014
32 A Simple Exercise Let p= It rained last night, q= he sprinklers came on last night, r= he lawn was wet this morning. ranslate each of the following into English: p = r p = r p q = CO Dr. Ungor Either the lawn wasn t wet this morning, or it rained last night, or the sprinklers came on last night. 14 uesday, January 7, 2014
33 he Exclusive Or Operator he binary exclusive-or operator (XOR) combines two propositions to form their logical exclusive or (exjunction?). p = I will earn an A in this course, q = I will drop this course, p q = I will either earn an A for this course, or I will drop it (but not both!) CO Dr. Ungor 15 uesday, January 7, 2014
34 Exclusive-Or ruth Note that p q means that p is true, or q is true, but not both! his operation is called exclusive or, able because it excludes the possibility that both p and q are true. and together are not universal. CO Dr. Ungor 16 uesday, January 7, 2014
35 Natural Language is Ambiguous Note that in English or is by itself ambiguous regarding the both case! Pat is a singer or Pat is a writer. - Pat is rich or Pat is poor. - Need context to disambiguate the meaning! or this class, assume or means inclusive. CO Dr. Ungor 17 uesday, January 7, 2014
36 Natural Language is Ambiguous Note that in English or is by itself ambiguous regarding the both case! Pat is a singer or Pat is a writer. - Pat is rich or Pat is poor. - Need context to disambiguate the meaning! or this class, assume or means inclusive. CO Dr. Ungor 17 uesday, January 7, 2014
37 Natural Language is Ambiguous Note that in English or is by itself ambiguous regarding the both case! Pat is a singer or Pat is a writer. - Pat is rich or Pat is poor. - Need context to disambiguate the meaning! or this class, assume or means inclusive. CO Dr. Ungor 17 uesday, January 7, 2014
38 he Implication Operator he implication p q states that p implies q. It is ALSE only in the case that p is RUE but q is ALSE. e.g., p= I am elected. q= axes will be lowered. p q = If I am elected, then taxes will be lowered (else it could go either way) CO Dr. Ungor 18 uesday, January 7, 2014
39 he Implication Operator erminology for the structure of implication p q p: Hypothesis (antecedent or premise) q: Conclusion (consequence) CO Dr. Ungor 19 uesday, January 7, 2014
40 Implication ruth able CO Dr. Ungor 20 uesday, January 7, 2014
41 Implication ruth able p q is false only when p is true but q is not true. CO Dr. Ungor 20 uesday, January 7, 2014
42 Implication ruth able p q is false only when p is true but q is not true. p q does not imply that p causes q! CO Dr. Ungor 20 uesday, January 7, 2014
43 Implication ruth able p q is false only when p is true but q is not true. p q does not imply that p causes q! p q does not imply that p or q are ever true! CO Dr. Ungor 20 uesday, January 7, 2014
44 Implication ruth able p q is false only when p is true but q is not true. p q does not imply that p causes q! p q does not imply that p or q are ever true! E.g. (1=0) pigs can fly is RUE! CO Dr. Ungor 20 uesday, January 7, 2014
45 Examples of Implications If this lecture ends, then the sun will rise tomorrow. rue or alse? If Monday is a day of the week, then I am a not teacher. rue or alse? If 1+1=6, then George passed the exam. rue or alse? If the moon is made of green cheese, then I am richer than Bill Gates. rue or alse? CO Dr. Ungor 21 uesday, January 7, 2014
46 Examples of Implications If this lecture ends, then the sun will rise tomorrow. rue or alse? If Monday is a day of the week, then I am a not teacher. rue or alse? If 1+1=6, then George passed the exam. rue or alse? If the moon is made of green cheese, then I am richer than Bill Gates. rue or alse? CO Dr. Ungor 21 uesday, January 7, 2014
47 Examples of Implications If this lecture ends, then the sun will rise tomorrow. rue or alse? If Monday is a day of the week, then I am a not teacher. rue or alse? If 1+1=6, then George passed the exam. rue or alse? If the moon is made of green cheese, then I am richer than Bill Gates. rue or alse? CO Dr. Ungor 21 uesday, January 7, 2014
48 Examples of Implications If this lecture ends, then the sun will rise tomorrow. rue or alse? If Monday is a day of the week, then I am a not teacher. rue or alse? If 1+1=6, then George passed the exam. rue or alse? If the moon is made of green cheese, then I am richer than Bill Gates. rue or alse? CO Dr. Ungor 21 uesday, January 7, 2014
49 Examples of Implications If this lecture ends, then the sun will rise tomorrow. rue or alse? If Monday is a day of the week, then I am a not teacher. rue or alse? If 1+1=6, then George passed the exam. rue or alse? If the moon is made of green cheese, then I am richer than Bill Gates. rue or alse? CO Dr. Ungor 21 uesday, January 7, 2014
50 Inverse, Converse, Contrapositive Some terminology: he inverse of p q is: p q he converse of p q is: q p. he contrapositive of p q is: q p. CO Dr. Ungor 22 uesday, January 7, 2014
51 Inverse, Converse, Contrapositive Some terminology: he inverse of p q is: p q he converse of p q is: q p. he contrapositive of p q is: q p. One of these has the same meaning (same truth table) as p q. Can you figure out which? CO Dr. Ungor 22 uesday, January 7, 2014
52 Inverse, Converse, Contrapositive Some terminology: he inverse of p q is: p q he converse of p q is: q p. he contrapositive of p q is: q p. One of these has the same meaning (same truth table) as p q. Can you figure out which? CO Dr. Ungor Contrapositive 22 uesday, January 7, 2014
53 How do we know for sure? Proving the equivalence of p q and its contrapositive using truth tables: CO Dr. Ungor 23 uesday, January 7, 2014
54 he biconditional operator CO Dr. Ungor 24 uesday, January 7, 2014
55 he biconditional operator he biconditional p q states that p is true if and only if (I) q is true. CO Dr. Ungor 24 uesday, January 7, 2014
56 he biconditional operator he biconditional p q states that p is true if and only if (I) q is true. It is RUE only when both p q and q p are RUE. CO Dr. Ungor 24 uesday, January 7, 2014
57 he biconditional operator he biconditional p q states that p is true if and only if (I) q is true. It is RUE only when both p q and q p are RUE. p = It is raining. CO Dr. Ungor 24 uesday, January 7, 2014
58 he biconditional operator he biconditional p q states that p is true if and only if (I) q is true. It is RUE only when both p q and q p are RUE. p = It is raining. q = he home team wins. CO Dr. Ungor 24 uesday, January 7, 2014
59 he biconditional operator he biconditional p q states that p is true if and only if (I) q is true. It is RUE only when both p q and q p are RUE. p = It is raining. q = he home team wins. p q = If and only if it is raining, the home team wins. CO Dr. Ungor 24 uesday, January 7, 2014
60 Biconditional ruth able CO Dr. Ungor 25 uesday, January 7, 2014
61 Biconditional ruth able p q means that p and q have the same truth value. CO Dr. Ungor 25 uesday, January 7, 2014
62 Biconditional ruth able p q means that p and q have the same truth value. Note this truth table is the exact opposite of ʼs! CO Dr. Ungor 25 uesday, January 7, 2014
63 Biconditional ruth able p q means that p and q have the same truth value. Note this truth table is the exact opposite of ʼs! p q means (p q) CO Dr. Ungor 25 uesday, January 7, 2014
64 Biconditional ruth able p q means that p and q have the same truth value. Note this truth table is the exact opposite of ʼs! p q means (p q) p q does not imply p and q are true, or cause each other. CO Dr. Ungor 25 uesday, January 7, 2014
65 Boolean Operations Summary We have seen 1 unary operator and 5 binary operators. CO Dr. Ungor 26 uesday, January 7, 2014
66 Precedence of Logical Operators Operator Precedence CO Dr. Ungor 27 uesday, January 7, 2014
67 Nested Propositional Expressions Use parentheses to group sub-expressions: I just saw my old friend, and either heʼs grown or Iʼve shrunk. = f (g s) (f g) s would mean something different f g s would be ambiguous By convention, takes precedence over both and. s f means ( s) f, not (s f) CO Dr. Ungor 28 uesday, January 7, 2014
68 Some Alternative Notations CO Dr. Ungor 29 uesday, January 7, 2014
69 Bits and Bit Operations A bit is a binary (base 2) digit: 0 or 1. By convention: 1 represents true ; Bits may be used to represent truth values. 0 represents false. Boolean algebra is like ordinary algebra except that variables stand for bits, + means or, and multiplication means and. CO Dr. Ungor 30 uesday, January 7, 2014
70 Bit Strings A Bit string of length n is an ordered series or sequence of n 0 bits. By convention, bit strings are written left to right: e.g. the first bit of is 1. When a bit string represents a base-2 number, by convention the first bit is the most significant bit. Ex =8+4+1=13. CO Dr. Ungor 31 uesday, January 7, 2014
71 Bitwise Operations Boolean operations can be extended to operate on bit strings as well as single bits. E.g.: Bit-wise OR Bit-wise AND Bit-wise XOR CO Dr. Ungor 32 uesday, January 7, 2014
72 Propositional Equivalences CO Dr. Entezari uesday, January 7, 2014
73 Propositional Equivalence wo syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). We call them equivalent. NEX: Learn about various equivalence rules or laws. Learn how to prove equivalences using symbolic derivations. CO Dr. Ungor 34 uesday, January 7, 2014
74 autologies and Contradictions CO Dr. Ungor 35 uesday, January 7, 2014
75 autologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! CO Dr. Ungor 35 uesday, January 7, 2014
76 autologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p p [What is its truth table?] CO Dr. Ungor 35 uesday, January 7, 2014
77 autologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p p [What is its truth table?] A contradiction is a comp. prop. that is false no matter what! Ex. p p [ruth table?] CO Dr. Ungor 35 uesday, January 7, 2014
78 autologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p p [What is its truth table?] A contradiction is a comp. prop. that is false no matter what! Ex. p p [ruth table?] Other comp. props. are contingencies. CO Dr. Ungor 35 uesday, January 7, 2014
79 Proving Equivalences CO Dr. Ungor 36 uesday, January 7, 2014
80 Proving Equivalences Compound propositions P and Q are logically equivalent to each other I P and Q contain the same truth values as each other in all rows of their truth tables. CO Dr. Ungor 36 uesday, January 7, 2014
81 Proving Equivalences Compound propositions P and Q are logically equivalent to each other I P and Q contain the same truth values as each other in all rows of their truth tables. Compound proposition P is logically equivalent to compound proposition Q, written P Q, I the compound proposition P Q is a tautology. CO Dr. Ungor 36 uesday, January 7, 2014
82 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
83 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
84 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
85 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
86 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
87 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
88 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
89 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
90 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
91 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
92 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
93 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
94 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
95 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
96 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
97 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
98 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
99 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
100 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
101 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
102 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
103 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
104 Proving Equivalence via ruth ables P Q Ex. Prove that } } p q ( p q). Ex Ex. Prove that p q ( p q). Ex. Prove that p q ( p q).. Prove thex. Prove that pq ( p q). at p q ( p q). CO Dr. Ungor 37 uesday, January 7, 2014
105 Equivalence Laws hese are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead. hey provide a pattern or template that can be used to match much more complicated propositions and to find equivalences for them. CO Dr. Ungor 38 uesday, January 7, 2014
106 Equivalence Laws - Examples Identity: p p p p Domination: p p Idempotent: p p p p p p Double negation: p p Commutative: p q q p p q q p Associative: (p q) r p (q r) (p q) r p (q r) CO Dr. Ungor 39 uesday, January 7, 2014
107 More Equivalence Laws Distributive: p (q r) (p q) (p r) p (q r) (p q) (p r) De Morgan s: " (p q) p q " (p q) p q CO Dr. Ungor 40 uesday, January 7, 2014
108 More Equivalence Laws Absorption: " p (p q) p p (p q) p rivial tautology/contradiction: p p p p CO Dr. Ungor 41 uesday, January 7, 2014
109 Defining Operators via Equivalences Using equivalences, we can define operators in terms of other operators. Implication: p q p q Biconditional: p q (p q) (q p) p q (p q) Exclusive or: p q (p q) (p q) p q (p q) (q p) CO Dr. Ungor 42 uesday, January 7, 2014
110 An Example Problem CO Dr. Ungor 43 uesday, January 7, 2014
111 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. CO Dr. Ungor 43 uesday, January 7, 2014
112 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. (p q) (p r) CO Dr. Ungor 43 uesday, January 7, 2014
113 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. (p q) (p r) [Expand definition of ] (p q) (p r) CO Dr. Ungor 43 uesday, January 7, 2014
114 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. (p q) (p r) [Expand definition of ] (p q) (p r) [Defn. of ] (p q) ((p r) (p r)) CO Dr. Ungor 43 uesday, January 7, 2014
115 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. (p q) (p r) [Expand definition of ] (p q) (p r) [Defn. of ] (p q) ((p r) (p r)) [DeMorganʼs Law] CO Dr. Ungor 43 uesday, January 7, 2014
116 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. (p q) (p r) [Expand definition of ] (p q) (p r) [Defn. of ] (p q) ((p r) (p r)) [DeMorganʼs Law] ( p q) ((p r) (p r)) CO Dr. Ungor 43 uesday, January 7, 2014
117 An Example Problem Check using a symbolic derivation whether (p q) (p r) p q r. (p q) (p r) [Expand definition of ] (p q) (p r) [Defn. of ] (p q) ((p r) (p r)) [DeMorganʼs Law] ( p q) ((p r) (p r)) CO Dr. Ungor 43 uesday, January 7, 2014
118 Example Continued... CO Dr. Ungor 44 uesday, January 7, 2014
119 Example Continued... ( p q) ((p r) (p r)) [ commutes] CO Dr. Ungor 44 uesday, January 7, 2014
120 Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] CO Dr. Ungor 44 uesday, January 7, 2014
121 Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] q ( p ((p r) (p r))) [distrib. over ] CO Dr. Ungor 44 uesday, January 7, 2014
122 Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] q ( p ((p r) (p r))) [distrib. over ] q ((( p (p r)) ( p (p r))) CO Dr. Ungor 44 uesday, January 7, 2014
123 Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] q ( p ((p r) (p r))) [distrib. over ] q ((( p (p r)) ( p (p r))) [assoc.] q ((( p p) r) ( p (p r))) CO Dr. Ungor 44 uesday, January 7, 2014
124 Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] q ( p ((p r) (p r))) [distrib. over ] q ((( p (p r)) ( p (p r))) [assoc.] q ((( p p) r) ( p (p r))) [trivial taut.] q (( r) ( p (p r))) CO Dr. Ungor 44 uesday, January 7, 2014
125 Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] q ( p ((p r) (p r))) [distrib. over ] q ((( p (p r)) ( p (p r))) [assoc.] q ((( p p) r) ( p (p r))) [trivial taut.] q (( r) ( p (p r))) [domination] q ( ( p (p r))) CO Dr. Ungor 44 uesday, January 7, 2014
126 CO Dr. Ungor Example Continued... ( p q) ((p r) (p r)) [ commutes] (q p) ((p r) (p r)) [ associative] q ( p ((p r) (p r))) [distrib. over ] q ((( p (p r)) ( p (p r))) [assoc.] q ((( p p) r) ( p (p r))) [trivial taut.] q (( r) ( p (p r))) [domination] q ( ( p (p r))) [identity] uesday, January 7, 2014 q ( p (p r)) cont. 44
127 End of Long Example CO Dr. Ungor 45 uesday, January 7, 2014
128 q ( p (p r)) End of Long Example CO Dr. Ungor 45 uesday, January 7, 2014
129 End of Long Example q ( p (p r)) [DeMorganʼs] q ( p ( p r)) CO Dr. Ungor 45 uesday, January 7, 2014
130 End of Long Example q ( p (p r)) [DeMorganʼs] q ( p ( p r)) [Assoc.] q (( p p) r) CO Dr. Ungor 45 uesday, January 7, 2014
131 End of Long Example q ( p (p r)) [DeMorganʼs] q ( p ( p r)) [Assoc.] q (( p p) r) [Idempotent] q ( p r) CO Dr. Ungor 45 uesday, January 7, 2014
132 End of Long Example q ( p (p r)) [DeMorganʼs] q ( p ( p r)) [Assoc.] q (( p p) r) [Idempotent] q ( p r) [Assoc.] (q p) r CO Dr. Ungor 45 uesday, January 7, 2014
133 End of Long Example q ( p (p r)) [DeMorganʼs] q ( p ( p r)) [Assoc.] q (( p p) r) [Idempotent] q ( p r) [Assoc.] (q p) r [Commut.] p q r CO Dr. Ungor 45 uesday, January 7, 2014
134 End of Long Example q ( p (p r)) [DeMorganʼs] q ( p ( p r)) [Assoc.] q (( p p) r) [Idempotent] q ( p r) [Assoc.] (q p) r [Commut.] p q r Q.E.D. (quod erat demonstrandum) CO Dr. Ungor 45 uesday, January 7, 2014
Propositional logic ( ): Review from Mat 1348
CSI 2101 / Winter 2008: Discrete Structures. Propositional logic ( 1.1-1.2): Review from Mat 1348 Dr. Nejib Zaguia - Winter 2008 1 Propositional logic: Review Mathematical Logic is a tool for working with
ICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
ARTIFICIAL INTELLIGENCE
ARTIFICIAL INTELLIGENCE LECTURE # 03 Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 1 Review of Last Lecture Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 2 Today s
Discrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
The Fundamentals of Logic. Foundations of Logic. Foundations of Logic. CMSC 302 Propositional Logic. In Rosen, ~98 slides, ~3 lectures
VCU, Department of Computer Science CMSC 302 Propositional Logic Vojislav Kecman The Fundamentals of Logic In Rosen, 1.1-1.4 ~98 slides, ~3 lectures All the PPT slides are based on MP Franck s and H Bingöl
CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)
CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of
CSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
Chapter Summary. Propositional Logic. Predicate Logic. Proofs. The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.
Chapter 1 Chapter Summary Propositional Logic The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.3) Predicate Logic The Language of Quantifiers (1.4) Logical Equivalences (1.4)
Proposition/Statement. Boolean Logic. Boolean variables. Logical operators: And. Logical operators: Not 9/3/13. Introduction to Logical Operators
Proposition/Statement Boolean Logic CS 231 Dianna Xu A proposition is either true or false but not both he sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 1 2 Boolean variables
Chapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of
Compound Propositions
Discrete Structures Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And 3. Or 4. Exclusive or 5. Implication 6. Biconditional Truth
Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
Example. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives
Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.
Sec$on Summary. Propositions Connectives. Truth Tables. Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional
Section 1.1 Sec$on Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional ruth ables 2 Proposi$ons A proposition is a declarative
Definition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
Propositional Logic. Propositional Equivalences. Agenda. Yutaka HATA, IEEE Fellow. Ch1.1 Propositional Logic. Ch1.2 Propositional Equivalences
Discrete Mathematic Chapter 1: Logic and Proof 1.1 Propositional Logic 1.2 Propositional Equivalences Yutaka HAA, IEEE ellow University of Hyogo, JAPAN his material was made by Dr. Patrick Chan School
Chapter 1, Section 1.1 Propositional Logic
Discrete Structures Chapter 1, Section 1.1 Propositional Logic These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6 th ed., by Kenneth H. Rosen, published
Chapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary! Propositional Logic! The Language of Propositions! Applications! Logical Equivalences! Predicate Logic! The Language
2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
Bits and Bit Operations. Today s topics
oday s topics Bits and Bit Operations opic #2 Bits Boolean algebra preview Propositional equivalences Predicate logic Reading: Sections 1.2-1.4 Slides taken from Prof. Michael rank A bit is a binary nary
Boolean Logic. CS 231 Dianna Xu
Boolean Logic CS 231 Dianna Xu 1 Proposition/Statement A proposition is either true or false but not both The sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 2 Boolean variables
1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
ICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
Propositional Logic 1
Propositional Logic 1 Section Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional Truth Tables 2 Propositions A proposition is
Discrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: Proposition logic MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 12, 2017 Outline 1 Propositions 2 Connectives
An Introduction to Logic 1.1 ~ 1.4 6/21/04 ~ 6/23/04
An Introduction to Logic 1.1 ~ 1.4 6/21/04 ~ 6/23/04 1 A Taste of Logic Logic puzzles (1) Knights and Knaves Knights: always tell the truth Knaves: always lie You encounter two people A and B. A says:
Section 1.1 Propositional Logic. proposition : true = T (or 1) or false = F (or 0) (binary logic) the moon is made of green cheese
Section 1.1 Propositional Logic proposition : true = T (or 1) or false = F (or 0) (binary logic) the moon is made of green cheese go to town! X - imperative What time is it? X - interrogative propositional
A statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
DISCRETE STRUCTURES WEEK5 LECTURE1
DISCRETE STRUCTURES WEEK5 LECTURE1 Let s get started with... Logic! Spring 2010 CPCS 222 - Discrete Structures 2 Logic Crucial for mathematical reasoning Important for program design Used for designing
Logic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:
Logic Logic is a discipline that studies the principles and methods used in correct reasoning It includes: A formal language for expressing statements. An inference mechanism (a collection of rules) to
Discrete Structures of Computer Science Propositional Logic I
Discrete Structures of Computer Science Propositional Logic I Gazihan Alankuş (Based on original slides by Brahim Hnich) July 26, 2012 1 Use of Logic 2 Statements 3 Logic Connectives 4 Truth Tables Use
Propositional Equivalence
Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:
Mathematical Logic Part One
Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going Propositional Logic (oday) Basic logical connectives. ruth tables. Logical
Overview. 1. Introduction to Propositional Logic. 2. Operations on Propositions. 3. Truth Tables. 4. Translating Sentences into Logical Expressions
Note 01 Propositional Logic 1 / 10-1 Overview 1. Introduction to Propositional Logic 2. Operations on Propositions 3. Truth Tables 4. Translating Sentences into Logical Expressions 5. Preview: Propositional
Propositional Logic. Spring Propositional Logic Spring / 32
Propositional Logic Spring 2016 Propositional Logic Spring 2016 1 / 32 Introduction Learning Outcomes for this Presentation Learning Outcomes... At the conclusion of this session, we will Define the elements
Introduction. Applications of discrete mathematics:
Introduction Applications of discrete mathematics: Formal Languages (computer languages) Compiler Design Data Structures Computability Automata Theory Algorithm Design Relational Database Theory Complexity
software design & management Gachon University Chulyun Kim
Gachon University Chulyun Kim 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs 3 1.1 Propositional Logic
HW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall
CSE 20 HW1 graded review form? HW2 released DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Translate sentences from English to propositional logic using appropriate
Truth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
Discrete Structures & Algorithms. Propositional Logic EECE 320 // UBC
Discrete Structures & Algorithms Propositional Logic EECE 320 // UBC 1 Review of last lecture Pancake sorting A problem with many applications Bracketing (bounding a function) Proving bounds for pancake
Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
Note: The area of logic that deals with propositions is called the propositional calculus or propositional logic.
Ch. 1.1 Logic Logic 1 Def. A Proposition is a statement that is either true or false. Example 1: Which of the following are propositions? Statement Proposition (yes or no) UHD is a University 1 + 3 = 0
1 The Foundations. 1.1 Logic. A proposition is a declarative sentence that is either true or false, but not both.
he oundations. Logic Propositions are building blocks of logic. A proposition is a declarative sentence that is either true or false, but not both. Example. Declarative sentences.. Ottawa is the capital
Section 1.1: Logical Form and Logical Equivalence
Section 1.1: Logical Form and Logical Equivalence An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of an argument is called the conclusion,
UNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
The Foundations: Logic and Proofs. Part I
The Foundations: Logic and Proofs Part I Chapter Summary Propositional Logic n The Language of Propositions n Applications n Logical Equivalences Predicate Logic n The Language of Quantifiers n Logical
Announcements. CS243: Discrete Structures. Propositional Logic II. Review. Operator Precedence. Operator Precedence, cont. Operator Precedence Example
Announcements CS243: Discrete Structures Propositional Logic II Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Tuesday 09/11 Weilin s Tuesday office hours are
Equivalence and Implication
Equivalence and Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February 7 8, 2018 Alice E. Fischer Laws of Logic... 1/33 1 Logical Equivalence Contradictions and Tautologies 2 3 4 Necessary
Functions. Lecture 4: Truth functions, evaluating compound statements. Arithmetic Functions. x y x+y
Lecture 4: ruth functions, evaluating compound statements 1. unctions, arithmetic functions, and truth functions 2. Definitions of truth functions unctions A function is something that takes inputs and
EECS 1028 M: Discrete Mathematics for Engineers
EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 26 Why Study Logic?
Introduction Propositional Logic. Discrete Mathematics Andrei Bulatov
Introduction Propositional Logic Discrete Mathematics Andrei Bulatov Discrete Mathematics Propositional Logic 2-2 What is Logic? Computer science is a mere continuation of logic by other means Georg Gottlob
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016 Propositions Proposition:
Discrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #2: Propositional Logic Based on materials developed by Dr. Adam Lee Today s Topic: Propositional
Propositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
13.3 Truth Tables for the Conditional and the Biconditional
ruthablesconditionalbiconditional.nb 1 13.3 ruth ables for the Conditional and the Biconditional Conditional Earlier, we mentioned that the statement preceding the conditional symbol is called the antecedent
Today s Topic: Propositional Logic
Today s Topic: Propositional Logic What is a proposition? Logical connectives and truth tables Translating between English and propositional logic Logic is the basis of all mathematical and analytical
Logic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations
Logic Propositions and logical operations Main concepts: propositions truth values propositional variables logical operations 1 Propositions and logical operations A proposition is the most basic element
AMTH140 Lecture 8. Symbolic Logic
AMTH140 Lecture 8 Slide 1 Symbolic Logic March 10, 2006 Reading: Lecture Notes 6.2, 6.3; Epp 1.1, 1.2 Logical Connectives Let p and q denote propositions, then: 1. p q is conjunction of p and q, meaning
Artificial Intelligence Knowledge Representation I
Artificial Intelligence Knowledge Representation I Agents that reason logically knowledge-based approach implement agents that know about their world and reason about possible courses of action needs to
n logical not (negation) n logical or (disjunction) n logical and (conjunction) n logical exclusive or n logical implication (conditional)
Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.6) TOPICS Propositional Logic Logical Operators Truth Tables Implication Logical Equivalence Inference Rules What you should know about propositional
CHAPTER 1. MATHEMATICAL LOGIC 1.1 Fundamentals of Mathematical Logic
CHAPER 1 MAHEMAICAL LOGIC 1.1 undamentals of Mathematical Logic Logic is commonly known as the science of reasoning. Some of the reasons to study logic are the following: At the hardware level the design
Section 3.1. Statements and Logical Connectives. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 3.1 Statements and Logical Connectives What You Will Learn Statements, quantifiers, and compound statements Statements involving the words not, and, or, if then, and if and only if 3.1-2 HISORY
1.1 Language and Logic
c Oksana Shatalov, Fall 2017 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
CSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Describe and use algorithms for integer operations based on their expansions Relate algorithms for integer
Logic and Proof. Aiichiro Nakano
Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science
CSE 240 Logic and Discrete Mathematics
CSE 240 Logic and Discrete Mathematics Instructor: odd Sproull Department of Computer Science and Engineering Washington University in St. Louis 1Extensible - CSE 240 Logic Networking and Discrete Platform
MAT2345 Discrete Math
Fall 2013 General Syllabus Schedule (note exam dates) Homework, Worksheets, Quizzes, and possibly Programs & Reports Academic Integrity Do Your Own Work Course Web Site: www.eiu.edu/~mathcs Course Overview
Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9
Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9 Typeset September 23, 2005 1 Statements or propositions Defn: A statement is an assertion
PROPOSITIONAL CALCULUS
PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and
CSCE 222 Discrete Structures for Computing. Propositional Logic. Dr. Hyunyoung Lee. !!!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Propositional Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Propositions A proposition is a declarative sentence that is either true or false
THE LOGIC OF COMPOUND STATEMENTS
THE LOGIC OF COMPOUND STATEMENTS All dogs have four legs. All tables have four legs. Therefore, all dogs are tables LOGIC Logic is a science of the necessary laws of thought, without which no employment
Why Learning Logic? Logic. Propositional Logic. Compound Propositions
Logic Objectives Propositions and compound propositions Negation, conjunction, disjunction, and exclusive or Implication and biconditional Logic equivalence and satisfiability Application of propositional
Chapter 1: The Logic of Compound Statements. January 7, 2008
Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive
What is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
1.1 Language and Logic
c Oksana Shatalov, Spring 2018 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
Introduction to Decision Sciences Lecture 2
Introduction to Decision Sciences Lecture 2 Andrew Nobel August 24, 2017 Compound Proposition A compound proposition is a combination of propositions using the basic operations. For example (p q) ( p)
LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, Propositional logic
LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, 2016 Propositional logic 1 Vocabulary of propositional logic Vocabulary (1) a. Propositional letters: p, q, r, s, t, p 1, q 1,..., p 2, q 2,...
Overview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR Last time Expert Systems and Ontologies oday Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof theory Natural
CSE 311: Foundations of Computing. Lecture 2: More Logic, Equivalence & Digital Circuits
CSE 311: Foundations of Computing Lecture 2: More Logic, Equivalence & Digital Circuits Last class: Some Connectives & Truth Tables Negation (not) p p T F F T Disjunction (or) p q p q T T T T F T F T T
Propositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
Topic 1: Propositional logic
Topic 1: Propositional logic Guy McCusker 1 1 University of Bath Logic! This lecture is about the simplest kind of mathematical logic: propositional calculus. We discuss propositions, which are statements
Mathematical Logic Part One
Mathematical Logic Part One An Important Question How do we formalize the logic we've been using in our proofs? Where We're Going Propositional Logic (oday) Basic logical connectives. ruth tables. Logical
PHI Propositional Logic Lecture 2. Truth Tables
PHI 103 - Propositional Logic Lecture 2 ruth ables ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value
DISCRETE MATHEMATICS BA202
TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION
Propositional Languages
Propositional Logic Propositional Languages A propositional signature is a set/sequence of primitive symbols, called proposition constants. Given a propositional signature, a propositional sentence is
THE LOGIC OF COMPOUND STATEMENTS
CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.1 Logical Form and Logical Equivalence Copyright Cengage Learning. All rights reserved. Logical Form
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement
Discrete Mathematics
Discrete Mathematics Discrete mathematics is devoted to the study of discrete or distinct unconnected objects. Classical mathematics deals with functions on real numbers. Real numbers form a continuous
Mathematical Logic Part One
Mathematical Logic Part One Announcements Problem Session tonight from 7:00 7:50 in 380-380X. Optional, but highly recommended! Problem Set 3 Checkpoint due right now. 2 Handouts Problem Set 3 Checkpoint
Logic. Def. A Proposition is a statement that is either true or false.
Logic Logic 1 Def. A Proposition is a statement that is either true or false. Examples: Which of the following are propositions? Statement Proposition (yes or no) If yes, then determine if it is true or
We last time we began introducing equivalency laws.
Monday, January 14 MAD2104 Discrete Math 1 Course website: www/mathfsuedu/~wooland/mad2104 Today we will continue in Course Notes Chapter 22 We last time we began introducing equivalency laws Today we
Logic and Proofs. (A brief summary)
Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations
Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
![Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.](/thumbs/87/97228443.jpg "Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.")
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
Logic and Propositional Calculus
CHAPTER 4 Logic and Propositional Calculus 4.1 INTRODUCTION Many algorithms and proofs use logical expressions such as: IF p THEN q or If p 1 AND p 2, THEN q 1 OR q 2 Therefore it is necessary to know
Introduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
LOGIC & PROPOSITIONAL EQUIVALENCE
KS091201 MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) LOGIC & PROPOSITIONAL EQUIVALENCE Discrete Math Team 2 -- KS091201 MD W-02 Outline Logic Proposition Propositional Variables Logical Operators Presedence
It is not the case that ϕ. p = It is not the case that it is snowing = It is not. r = It is not the case that Mary will go to the party =
Introduction to Propositional Logic Propositional Logic (PL) is a logical system that is built around the two values TRUE and FALSE, called the TRUTH VALUES. true = 1; false = 0 1. Syntax of Propositional
Propositions. Frequently, we will use the word statement instead of proposition.
Propositional Logic Propositions A proposition is a declaration of fact that is either true or false, but not both. Examples and non-examples: One plus two equals four (proposition) Mozart is the greatest