CSE 20: Discrete Mathematics
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1 Spring 2018
2 Last Time Welcome / Introduction to Propositional Logic Card Puzzle / Bar Puzzle Same Problem Same Logic Same Answer Logic is a science of the necessary laws of thought (Kant, 1785) Logic allow to determine whether a statement is true or not based on its form, rather than what you already know about the subject matter. This is very useful when answering new questions, about new problems, when intuition does not help.
3 Today More serious look at propositional logic Propositions Logical connectives: negation, conjunction, disjunction, implication Compound statements Truth tables Reading: Chap. 1.1, 1.2, 1.3
4 Propositions A proposition is a declarative sentence that is either true or false. Examples: Rome is the capital of Italy San Francisco is the capital of California = 2 2π = 6.28 Prof. Micciancio has a thick Italian accent
5 Propositions A proposition is a declarative sentence that is either true or false. Examples: Rome is the capital of Italy San Francisco is the capital of California = 2 2π = 6.28 Prof. Micciancio has a thick Italian accent Some are true, some are false, but they are all legitimate propositions. True: T 1 yes on False: F 0 no off
6 More Propositions? Consider the following sentences. Only one of them is a proposition. (A) Answer this question! (B) What time is it? (C) 4 + x = 42 (D) 2 3 > 8 (E) This statement is not true
7 More Propositions? Consider the following sentences. Only one of them is a proposition. (A) Answer this question! (B) What time is it? (C) 4 + x = 42 (D) 2 3 > 8 (E) This statement is not true Correct answer: (D)
8 Compound Propositions Propositions obtained combining smaller propositions using logical connectives. A = My computer has at least 4GB of RAM B = My computer has at least 1TB of SSD storage
9 Compound Propositions Propositions obtained combining smaller propositions using logical connectives. A = My computer has at least 4GB of RAM B = My computer has at least 1TB of SSD storage Compound statement: C = A and B My computer has at least 4GB of RAM and my computer has at least 1TB of SSD storage My computer has at least 4GB of RAM and 1TB of SSD storage
10 Compound Propositions Propositions obtained combining smaller propositions using logical connectives. A = My computer has at least 4GB of RAM B = My computer has at least 1TB of SSD storage Compound statement: C = A and B My computer has at least 4GB of RAM and my computer has at least 1TB of SSD storage My computer has at least 4GB of RAM and 1TB of SSD storage Compound statement: D = not A It is not the case that my computer has at least 4GB of RAM My computer has less than 4GB of RAM Propositional logic gives precise meaning to the words and, not, etc.
11 Negation (not) A: Proposition Notation: not A, A, A Meaning: It is not the case that A is true. A A T F F T Truth Table: For every value of A, specifies the value of A
12 Conjunction (and) A, B: Propositions Notation: A and B, A B, A&B Meaning: A and B are both true A B A B F F F F T F T F F T T T Truth Table: For every value of A and B, specifies the value of A B
13 Disjunction (or) A, B: Propositions Notation: A or B, A B, A B Meaning: at least one of A or B is true A B A B F F F F T T T F T T T T Note: this is an inclusive or. A B means A or B or both are true.
14 Exlusive OR (xor) A B: A or B but not both A B A B F F F F T T T F T T T F
15 Exlusive OR (xor) A B: A or B but not both A B A B F F F F T T T F T T T F Same as A B Romans had two different words for or : vel : inclusive or aut : exlusive or
16 ... but It is not raining, but it is very cold Can you define but as a logical connective? A: It is not raining B: It is very cold A B A but B (A) (B) (C) (D) F F F F T F T F??? F T F T T T??? F F T T
17 ... but It is not raining, but it is very cold Can you define but as a logical connective? A: It is not raining B: It is very cold A B A but B (A) (B) (C) (D) F F F F T F T F??? F T F T T T??? F F T T Answer: (C) A but B is the same as A B
18 ... if If it rains, I will stay home A: It rains B: I will stay home A B if A then B (A) (B) (C) (D) T T T T F F F T??? F T F T F F??? F F T T
19 ... if If it rains, I will stay home A: It rains B: I will stay home A B if A then B (A) (B) (C) (D) T T T T F F F T??? F T F T F F??? F F T T Answer: (D)
20 Implication (if) A, B: Propositions Notation: if A then B, A implies B, A B, A = B Meaning: if A is true, then B is also true A B A B F F T F T T T F F T T T A B is true, except when A is true and B is false
21 ... unless I will get an A in CSE20, unless I get sick A = I will get an A in CSE20 B = I get sick A B A unless B F F??? F T??? T F??? T T??? A unless B has the same meaning as (A) A B, (B) A B, (C) A B, (D) A B,
22 ... unless I will get an A in CSE20, unless I get sick A = I will get an A in CSE20 B = I get sick A B A unless B F F??? F T??? T F??? T T??? A unless B has the same meaning as (A) A B, (B) A B, (C) A B, (D) A B, Equivalent to ( A B): false if don t get sick, and still don t get an A A B: either I get an A or I get sick (or both)
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