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Show that the proposition ((p → q) ^ (q → r)) → (p –→r) is a tautology.

Show that the proposition ((p → q) ^ (q → r)) → (p –→r) is a tautology.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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a) Show that the proposition ((p –→ q) ^ (q → r)) → (p → r) is a tautology. b) Let Q(x, y) be the statement x? = y? on the domain Z of all integers. What are the truth values for the following propositions? Explain your reasoning. (i) Q(1,–1) (ii) Jx 3y (x # y ^ Q(x, y)) (iii) Vy Jr Q(x, y) c) For the following table of a Boolean function of the Boolean variables x, y, and 2, write down a formula for F(x, y, z) in disjunctive normal form (i.e., as a 'sum of products'). Then minimise the resulting expression, and draw the corresponding minimal Boolean circuit. x |y | z | F(x, y, z) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Transcribed Image Text:a) Show that the proposition ((p –→ q) ^ (q → r)) → (p → r) is a tautology. b) Let Q(x, y) be the statement x? = y? on the domain Z of all integers. What are the truth values for the following propositions? Explain your reasoning. (i) Q(1,–1) (ii) Jx 3y (x # y ^ Q(x, y)) (iii) Vy Jr Q(x, y) c) For the following table of a Boolean function of the Boolean variables x, y, and 2, write down a formula for F(x, y, z) in disjunctive normal form (i.e., as a 'sum of products'). Then minimise the resulting expression, and draw the corresponding minimal Boolean circuit. x |y | z | F(x, y, z) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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