QUESTION 1

(a) If [tex]\( p \)[/tex] stands for "Beatrice is beautiful", [tex]\( q \)[/tex] stands for "She is very neat", and [tex]\( r \)[/tex] stands for "She is highly intelligent":

(i) Give the verbal translation and construct the truth tables for the following compound statements:
1. [tex]\(\sim(\sim p \vee \sim q \wedge \sim r)\)[/tex]
2. [tex]\(( p \wedge q ) \downarrow r\)[/tex]
3. [tex]\(p \Rightarrow (q \vee \sim r)\)[/tex]

(6 marks)

(b) Prove the following statements:

(i) [tex]\(\sim[( p \wedge q ) \Rightarrow ( p \vee q )]\)[/tex] is a contradiction
(ii) [tex]\(\sim[( p \wedge q ) \wedge \sim( p \vee q )]\)[/tex] is a tautology
(iii) [tex]\(\sim[( p \Rightarrow q ) \wedge ( q \Rightarrow r ) \Rightarrow ( p = r )]\)[/tex] is a contradiction
(iv) [tex]\(\sim[( p \wedge q ) \wedge \sim p ]\)[/tex] is a tautology

(c) (i) Prove that [tex]\(p \Rightarrow q\)[/tex] is equivalent to [tex]\(( p \wedge q ) \vee (\sim p \wedge \sim q )\)[/tex]

(ii) Are the following statements equivalent?
- "It is false that she is both beautiful and intelligent"
- "If she is beautiful, she cannot be intelligent"

(iii) Prove that the following argument is invalid:
- If the sun is shining, he will play tennis.
- The sun was not shining.
- Therefore, he did not play tennis.

(5 [tex]\(\frac{1}{2}\)[/tex] marks)