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hw1_sol - EE 369 Homework 2 Solutions 4 points per...

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EE 369 Homework 2 Solutions 4 points per problem (except 8 points for E5) Section 1.1 10 (a) B ^ D (b) A ^ D (c) D --> (B V C) (d) B' ^ A (e) D --> C 13 Let H, K, A be the following statements: H : The horse is fresh K : The knight wins A : The armor is strong (a) H --> K (b) K --> (H ^ A) (c) K --> H (d) K <--> A (e) (A v H) --> K 17 (a),(b) please see the back of the textbook (c) A | B | A' | B' | (A' v B') | (A' v B')' | (A ^ (A' v B')') ------------------------------------------------------------- T | T | F | F | F | T | T T | F | F | T | T | F | F F | T | T | F | T | F | F F | F | T | T | T | F | F (d) A | B | (A ^ B) | A' | (A ^ B) --> A' --------------------------------------- T | T | T | F | F T | F | F | F | T F | T | F | T | T F | F | F | T | T (e) A | B | C | A-->B | AvC | BvC | [(AvC)-->(BvC)] | (A-->B) --> [(AvC)-->(BvC)] ------------------------------------------------------------------------------ T | T | T | T | T | T | T | T T | T | F | T | T | T | T | T T | F | T | F | T | T | T | T T | F | F | F | T | F | F | T F | T | T | T | T | T | T | T F | T | F | T | F | T | T | T F | F | T | T | T | T | T | T F | F | F | T | F | F | T | T is a tautology
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(f) A | B | B-->A | A-->(B-->A) ----------------------------- T | T | T | T T | F | T | T F | T | F | T F | F | T | T is a tautology (g) A | B | (A ^ B) | B' | A' | (B' v A') | (A ^ B) <--> (B' v A') ---------------------------------------------------------------- T | T | T | F | F | F | F T | F | F | T | F | T | F F | T | F | F | T | T | F F | F | F | T | T | T | F is a contradiction (h) A | B | B' | (A v B' ) | (A ^ B) | (A ^ B)' | (A v B' ) ^ (A ^ B)' -------------------------------------------------------------------- T | T | F | T | T | F | F T | F | T | T | F | T | T F | T | F | F | F | T | F F | F | T | T | F | T | T (i) A | B | C | AVB | C'| [(AvB)^C'] | A'| A'vC | [(AvB)^C']-->A'vC ------------------------------------------------------------------------------ T | T | T | T | F | F | F | T | T T | T | F | T | T | T | F | F | F T | F | T | T | F | F | F | T | T T | F | F | T | T | T | F | F | F F | T | T | T | F | F | T | T | T F | T | F | T | T | T | T | T | T F | F | T | F | F | F | T | T | T F | F | F | F | T | F | T | T | T 33 (a) We have to show that A-->B and A^B can be replaced by equivalent wffs that use only v and ' in any compound wff. We can show that A^B <==> (A')'^(B')' (see truth table below) <==> (A'vB')' (from De Morgan's law) (P <==> Q means P and Q are equivalent wffs as defined in page 8 of the text book), and A-->B <==> A'vB (see truth table below), hence any A-->B and A^B can be replaced by equivalent wffs that use only v and ' in any compound wff. A | B | A-->B | A' | B' | A'vB | (A')' | (B')' | A^B | (A')'^(B')' ------------------------------------------------------------------- T | T | T | F | F | T | T | T | T | T T | F | F | F | T | F | T | F | F | F F | T | T | T | F | T | F | T | F | F F | F | T | T | T | T | F | F | F | F
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(b) We have to show that AvB and A^B can be replaced by equivalent wffs that use only --> and ' in any compound wff.
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