Question

3. Use a truth table to show that $p \rightarrow q \equiv (p \land r) \rightarrow (q \land r)$ holds. [12 marks] 4. Prove the following, using the semantic equivalences given on the slides of logic lecture 3. In each step apply only one rule once and state the name of the rule. (a) $\neg p \leftrightarrow q \equiv p \leftrightarrow \neg q$ (b) $(p \land q) \rightarrow r \equiv (p \rightarrow r) \lor (q \rightarrow r)$ [18 marks] [18 marks]

          3. Use a truth table to show that $p \rightarrow q \equiv (p \land r) \rightarrow (q \land r)$ holds. [12 marks]
4. Prove the following, using the semantic equivalences given on the slides of logic lecture 3. In each step apply only one rule once and state the name of the rule.
(a) $\neg p \leftrightarrow q \equiv p \leftrightarrow \neg q$
(b) $(p \land q) \rightarrow r \equiv (p \rightarrow r) \lor (q \rightarrow r)$ [18 marks] [18 marks]

3. Use a truth table to show that p → q ≡ (p r) → (q r) holds. [12 marks]
4. Prove the following, using the semantic equivalences given on the slides of logic lecture 3. In each step apply only one rule once and state the name of the rule.
(a) p ↔ q ≡ p ↔ q
(b) (p q) → r ≡ (p → r) (q → r) [18 marks] [18 marks]

Added by Krystal L.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Brooke Bussoletti

Step 1

To do this, we will create a truth table with columns for p, q, r, p ∧ ¬q, ¬(p ∧ q), p ∧ ¬q ∨ r, and p ∧ ¬q → r ↔ ¬(p ∧ q) ∨ r. Here is the truth table: | p | q | r | p ∧ ¬q | ¬(p ∧ q) | p ∧ ¬q ∨ r | p ∧ ¬q → r ↔ ¬(p ∧ q) ∨ r Show more…

Show all steps

Thanks for your feedback!

3. Use a truth table to show that p âˆ§ Â¬q â†’ r â†” Â¬(p âˆ§ q) âˆ¨ r holds. [12 marks] 4. Prove the following, using the semantic equivalences given on the slides of logic lecture 3. In each step, apply only one rule once and state the name of the rule. a âˆ§ (p â†’ q) âˆ§ (p â†’ Â¬q) â†’ (p âˆ§ Â¬q) â†’ (p â†’ q) âˆ§ (p â†’ Â¬q) [18 marks] [18 marks]