CSE 20 DISCRETE MATH WINTER
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1 CSE 20 DISCRETE MATH WINTER
2 Reminders Exam 1 in one week One note sheet ok Review sessions Saturday / Sunday Assigned seats: seat map on Piazza shortly Homework 3 due Friday Office hours
3 Today's learning goals Relate boolean operations to applications Combinatorial circuits Logic puzzles Identify when and prove that a statement is a tautology or contradiction Prove propositional equivalences using truth tables Prove propositional equivalences using other known equivalences, e.g. DeMorgan s laws Double negation laws Distributive laws, etc. Compute the CNF and DNF of a given compound proposition. Identify when and prove that a statement is satisfiable or unsatisfiable, and when a set of statements is consistent or inconsistent.
4 Tautology and contradiction Rosen p. 25 Tautology: compound proposition that is always T Contradiction: compound proposition that is always F p q F T T T F T T F F T F T F T F F F T
5 System specification + consistency Rosen p. 23 #11 The router can send packets to the edge system only if it supports the new address space. For the router to support the new address space, it is necessary that the latest software release be installed. The router can send packets to the edge system if the latest software release is installed. The router does not support the new address space.
6 System specification + consistency Rosen p. 23 #11 p only if q. For q, it is necessary that r. p if r. NOT q.
7 System specification + consistency Rosen p. 23 #11 p only if q. For q, it is necessary that r. p if r. NOT q.
8 System specification + consistency Rosen p. 23 #11 p q r T T T T T F T F T T F F F T T F T F F F T F F F How many rows will be set to T in 4 th column? A. 1 B. 2 C. 4 D. 8 E. None of the above Remote frequency: CA To change your remote frequency 1. Press and hold power button until flashing 2. Enter two-letter code 3. Checkmark / green light indicates success
9 System specification + consistency Rosen p. 23 #11 p q r T T T T T T F T T F T F T F T F T F T T T T F F F T T T F T T T T F F F T F T F T F F F T T T F T F F F T T T T
10 System specification + consistency Rosen p. 18 p q r What property T of the truth T table T T T T F guarantees consistency? T T F T F T F A. All output rows T are T. F T F T T T B. All output rows T have F both T and F F. F T T T C. At least one row has all outputs T. D. At least one F row has T all outputs T F. T T F F E. None of the F above T F T F T F F F T T T F T F F F T T T T System specifications are consistent if they do not contain conflicting requirements
11 System specification + consistency Rosen p. 18 p q r T T T T T T F T T F T F T F T F T F T T T T F F F T T T F T T T T F F F T F T F T F F F T T T F T F F F T T T T System specifications are consistent if they do not contain conflicting requirements
12 System specification + consistency Rosen p. 23 #11 p is false: The router cannot send packets to the edge system. q is false: The router does not support the new address space. r is false: The latest software release is not installed. The router can send packets to the edge system only if it supports the new address space. For the router to support the new address space, it is necessary that the latest software release be installed. The router can send packets to the edge system if the latest software release is installed.the router does not support the new address space.
13 Circuits What function is implemented by each of these circuits?
14 Circuits A B 1 1? 1 0? 0 1? 0 0? What function is implemented by each of these circuits?
15 Circuits A B
16 Circuits Which of these circuits implement logically equivalent functions? A. C. B. D. E. None of the above / more than one of the above.
17 Logical equivalences Rosen p. 25 Compound propositions that have the same truth values in all possible cases are logically equivalent, denoted. p q T T? T F? F T? F F?
18 De Morgan Laws Rosen p. 26 Replacing the main connective!
19 (Some) Useful equivalences Rosen p equivalences listed in book! Can replace p and q with any (compound) proposition
20 (Some) Useful equivalences Rosen p For constructing (minimal) circuits with specified gates only NOTs? only ANDs?. 32 equivalences listed in book! Can replace p and q with any (compound) proposition
21 (Some) Useful equivalences Rosen p For simplying and evaluating complicated compound propositions Remove parentheses? Reduce subexpressions to simpler ones. 32 equivalences listed in book! Can replace p and q with any (compound) proposition
22 (Some) Useful equivalences Rosen p For devising proofs of statements Translate using existing logical structure. Try to apply known proof strategy. Rewrite in equivalent way to apply additional proof strategies. (more on this later). 32 equivalences listed in book! Can replace p and q with any (compound) proposition
23 Other laws of equivalence Rosen p Any compound proposition can be translated to one using A. only ANDs and ORs. B. only ANDs and NOTs. C. only IFs. D. only NOTs. E. None of the above
24 Other laws of equivalence Rosen p. 35 #42-53 Any compound proposition can be translated to one using A. only ANDs and ORs. B. only ANDs and NOTs. C. only IFs. D. only NOTs. E. None of the above Functionally complete collection of connectives.
25 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out, starting with connectives applied to basic propositions / propositional variables.
26 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing.
27 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with
28 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with
29 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with If the connective is IF..THEN: replace with
30 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with If the connective is IF..THEN: replace with
31 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with If the connective is IF..THEN: replace If the connective is IFF: replace with with
32 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with If the connective is IF..THEN: replace If the connective is IFF: replace with with
33 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with If the connective is IF..THEN: replace If the connective is IFF: replace with If the connective is XOR: replace with with
34 Functionally complete set of connectives Rosen p. 35 #42-53 Rewriting compound propositions using only NOT, AND 1. Work from the inside out 2. For each connective, replace it with an equivalent form that uses only NOT, AND: If the connective is NOT or AND, do nothing. If the connective is OR: replace with If the connective is IF..THEN: replace If the connective is IFF: replace with If the connective is XOR: replace with with
35 Functionally complete set of connectives Rosen p. 35 #42-53 Example: express as a logically equivalent compound proposition that only uses ANDs and NOTs.
36 Functionally complete set of connectives Rosen p. 35 #42-53 Example: express as a logically equivalent compound proposition that only uses ANDs and NOTs. Use to rewrite intermediate step:
37 Functionally complete set of connectives Rosen p. 35 #42-53 Example: express as a logically equivalent compound proposition that only uses ANDs and NOTs. Use to rewrite intermediate step: Use to rewrite: Simplify double negation:
38 Sample questions Given compound proposition What is its main connective? Is it a tautology? a contradiction? Construct its truth table. Construct a combinatorial circuit using. that produces the values of this proposition as output. Is it logically equivalent to.? Find its negation and "simplify". Find a logically equivalent compound proposition that
39 Going backwards Given compound proposition, use Truth tables Logical equivalences to compute truth values. Reverse? Given truth table settings, want compound proposition with that output.
40 CNF and DNF Rosen p. 35 #42-53 Conjunctive normal form: AND of ORs (of variables or their negations). Disjunctive normal form: OR of ANDs (of variables or their negations). Which of the following is in CNF? A. B. C. D. E. More than one of the above.
41 Reverse-engineering p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
42 Reverse-engineering Approach 1: classify rows based on one variable p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
43 Reverse-engineering Approach 2: algorithmically convert to normal form p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
44 Reverse-engineering Approach 2: algorithmically convert to normal form DNF: when is output T? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F LAND IN THESE ROWS!
45 Reverse-engineering Approach 2: algorithmically convert to normal form DNF: when is output T? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
46 Reverse-engineering Approach 2: algorithmically convert to normal form DNF: when is output T? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
47 Reverse-engineering Approach 2: algorithmically convert to normal form DNF: when is output T? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
48 Reverse-engineering Approach 2: algorithmically convert to normal form DNF: when is output T? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F Why is it ok to ignore the F rows? A. It's not, we need to add clauses from them too. B. If we don't specify a value it's automatically F. C. The definition "and" will set the output to "F" if the T row conditions aren't met. D. More than one of the above. E. None of the above.
49 Reverse-engineering Approach 2: algorithmically convert to normal form CNF: when is output F? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F AVOID THESE ROWS!
50 Reverse-engineering Approach 2: algorithmically convert to normal form CNF: when is output F? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F AVOID THESE ROWS!
51 Reverse-engineering Approach 2: algorithmically convert to normal form CNF: when is output F? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
52 Reverse-engineering Approach 2: algorithmically convert to normal form CNF: when is output F? p q r? T T T T T T F T T F T F T F F T F T T F F T F F F F T T F F F F
53 Normal forms Rosen p. 35 #42-53 Compound proposition Proposition in normal form Added benefit: If want to reduce connectives further to prove a new collection of connectives is functionally complete, only need to consider those used in normal form.
54 Reminders Exam 1 in one week One note sheet ok Review sessions Saturday / Sunday Assigned seats: seat map on Piazza shortly Homework 3 due Friday Office hours
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