Question

3. Without using the truth table prove the following equivalences. (a) \( p \rightarrow(q \vee r) \equiv(p \rightarrow q) \vee(p \rightarrow r) \). (b) \( (\neg p \rightarrow \neg q) \rightarrow(q \rightarrow p) \equiv T \) 4. Prove that if \( x \) is irrational, then \( 1 / x \) is irrational.

          3. Without using the truth table prove the following equivalences.
(a) \( p \rightarrow(q \vee r) \equiv(p \rightarrow q) \vee(p \rightarrow r) \).
(b) \( (\neg p \rightarrow \neg q) \rightarrow(q \rightarrow p) \equiv T \)
4. Prove that if \( x \) is irrational, then \( 1 / x \) is irrational.

3. Without using the truth table prove the following equivalences.
(a) p →(q ∨ r) ≡(p → q) ∨(p → r).
(b) ( p → q) →(q → p) ≡ T
4. Prove that if x is irrational, then 1 / x is irrational.

Added by James A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert William Mead

Step 1

- Use the definition of implication: \( p \rightarrow x \equiv \neg p \vee x \). Show more…

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3. Without using the truth table prove the following equivalences. (a) \( p \rightarrow(q \vee r) \equiv(p \rightarrow q) \vee(p \rightarrow r) \). (b) \( (\neg p \rightarrow \neg q) \rightarrow(q \rightarrow p) \equiv T \) 4. Prove that if \( x \) is irrational, then \( 1 / x \) is irrational.