CSE 20 DISCRETE MATH WINTER
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1 CSE 20 DISCRETE MATH WINTER
2 Reminders Homework 3 due Sunday at noon Exam 1 in one week One note card can be used. Bring photo ID. Review sessions Thursday in class Saturday January 28 11am-12:50pm PETER108 Recommended textbook qs Monday January 30 8pm-9:50pm SOLIS107 Exam Practice Sheet Assigned seats: seat map on Piazza shortly Office hours
3 Today's learning goals 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. Relate boolean operations to applications Combinatorial circuits Logic puzzles
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 Which of the following is a tautology? A. B. C. D. E. Clicker frequency: AC
5 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? Notice: A and B are logically equivalent iff Aß à B is a tautology
6 De Morgan Laws Rosen p. 26 Replacing the main connective!
7 (Some) Useful equivalences Rosen p equivalences listed in book! Can replace p and q with any (compound) proposition
8 (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
9 (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
10 (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
11 Sample equivalence proof Prove that is logically equivalent to Are these compound propositions logically equivalent to? You'll decide in HW3!
12 Other laws of equivalence Rosen p Any compound proposition can be translated to one using A. only ANDs. B. only ORs. C. only IFs. D. only NOTs. E. None of the above
13 Other laws of equivalence Rosen p. 35 #42-53 Any compound proposition can be translated to one using A. only ANDs. B. only ORs. C. only IFs. D. only NOTs. E. None of the above Functionally complete collection of connectives.
14 Functionally complete set of connectives Rosen p. 35 #42-53 Claim: The connectives AND, NOT are functionally complete. Any compound proposition can be rewritten as a logically equivalent one that only has the operators AND, NOT
15 Functionally complete set of connectives Rosen p. 35 #42-53 Claim: The connectives AND, NOT are functionally complete. Any compound proposition can be rewritten as a logically equivalent one that only has the operators AND, NOT Example:
16 Functionally complete set of connectives Rosen p. 35 #42-53 Claim: The connectives AND, NOT are functionally complete. Any compound proposition can be rewritten as a logically equivalent one that only has the operators AND, NOT Example:
17 Functionally complete set of connectives Rosen p. 35 #42-53 Claim: The connectives AND, NOT are functionally complete. Any compound proposition can be rewritten as a logically equivalent one that only has the operators AND, NOT Example:
18 Functionally complete set of connectives Rosen p. 35 #42-53 Claim: The connectives AND, NOT are functionally complete. Any compound proposition can be rewritten as a logically equivalent one that only has the operators AND, NOT Example:
19 Functionally complete set of connectives Rosen p. 35 #42-53 Claim: The connectives AND, NOT are functionally complete. Any compound proposition can be rewritten as a logically equivalent one that only has the operators AND, NOT Example:
20 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
21 Functionally complete set of connectives Rosen p. 35 #42-53 Example: express as a logically equivalent compound proposition that only uses ANDs and NOTs.
22 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:
23 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 and use associativity:
24 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
25 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. E.g. Think back to HW2 Q5
26 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. In this context, the propositional variable A can be interpreted as an AND (of itself), and as an OR (of itself)
27 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
28 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
29 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
30 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!
31 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
32 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
33 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
34 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
35 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!
36 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!
37 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
38 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
39 Payoff Any output column of a truth table (assignment of T/F to each combination of T/F input values) can be realized as a compound proposition. The collection is functionally complete.
40 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.
41 Puzzle: Muddy children Rosen Example 8, page 20 Two kids play in the mud outside. When they are done, their parent says at least one of you has a muddy forehead. The children can t see themselves but can see each other. If the older child sees mud on the younger child's forehead A. The older child knows both of them have muddy foreheads. B. The older child knows the younger child has a muddy forehead. C. The older child knows that they themselves have a muddy forehead. D. None of the above they can t be sure. E. I don t know.
42 Puzzle: Muddy children Rosen Example 8, page 20 Two kids play in the mud outside. When they are done, their parent says at least one of you has a muddy forehead. The children can t see themselves but can see each other. If the older child (even being a perfect reasoner) doesn't have enough information to tell whether s/he has mud on their forehead A. The younger child knows the older one has muddy forehead. B. The younger child knows the older one has no mud. C. The younger child knows s/he himself has no mud. D. None of the above they can t be sure. E. I don't know.
43 Reminders Homework 3 due Sunday at noon Exam 1 in one week One note card can be used. Bring IDs. Review sessions In class Thursday On the weekend TBA (if we get a lecture hall) Monday January 30 8pm-9:50pm Assigned seats: seat map on Piazza shortly Office hours listed on class calendar
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