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33. a. Show that for the Sheffer stroke |, PAQ = (P|Q)|(P|Q). b. Use the results of Example 2.4.7 and part (a) above to write P^(~Q v R) using only Sheffer strokes.

33. a. Show that for the Sheffer stroke |, PAQ = (P|Q)|(P|Q). b. Use the results of Example 2.4.7 and part (a) above to write P^(~Q v R) using only Sheffer strokes.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 87E
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Discrete math. Q33 Please.

Example 2.4.7 Rewriting Expressions Using the Sheffer Stroke Use Theorem 2.1.1 and the definition of Sheffer stroke to show that a. ~P = P|P and b. PVQ = (P|P)|(Q|Q). Solution a. ~P=~(P ^ P) = P|P by the idempotent law for A by definition of l. b. PV Q=~(~(P v Q)) = ~(~P ^~Q) = ~((P|P) ^ (Q|Q)) (P|P)|(Q|Q) by the double negative law by De Morgan's laws by part (a) by definition of |.
Transcribed Image Text:Example 2.4.7 Rewriting Expressions Using the Sheffer Stroke Use Theorem 2.1.1 and the definition of Sheffer stroke to show that a. ~P = P|P and b. PVQ = (P|P)|(Q|Q). Solution a. ~P=~(P ^ P) = P|P by the idempotent law for A by definition of l. b. PV Q=~(~(P v Q)) = ~(~P ^~Q) = ~((P|P) ^ (Q|Q)) (P|P)|(Q|Q) by the double negative law by De Morgan's laws by part (a) by definition of |.
For the circuits corresponding to the Boolean expressions in each of 30 and 31 there is an equivalent circuit with at most two logic gates. Find such a circuit. 30. (PA Q) v (~P^Q) v (~P ^~Q) 31. (~P ^~Q) v (~P^ Q) v (P^~Q) 32. The Boolean expression for the circuit in Example 2.4.5 is (P^Q^ R) V (P ^~Q^ R) V (P ^ ~Q ^ ~R) (a disjunctive normal form). Find a circuit with at most three logic gates that is equivalent to this circuit. 33. a. Show that for the Sheffer stroke , PAQ = (P|Q)|(P|Q). b. Use the results of Example 2.4.7 and part (a) above to write P^(~Qv R) using only Sheffer strokes.
Transcribed Image Text:For the circuits corresponding to the Boolean expressions in each of 30 and 31 there is an equivalent circuit with at most two logic gates. Find such a circuit. 30. (PA Q) v (~P^Q) v (~P ^~Q) 31. (~P ^~Q) v (~P^ Q) v (P^~Q) 32. The Boolean expression for the circuit in Example 2.4.5 is (P^Q^ R) V (P ^~Q^ R) V (P ^ ~Q ^ ~R) (a disjunctive normal form). Find a circuit with at most three logic gates that is equivalent to this circuit. 33. a. Show that for the Sheffer stroke , PAQ = (P|Q)|(P|Q). b. Use the results of Example 2.4.7 and part (a) above to write P^(~Qv R) using only Sheffer strokes.
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