a) By use of truth tables, establish whether the statements given below are a tantology or contradiction (i) (pwedge q)vee (neg pvee (pwedge neg q) (5 marks) (ii) . (pwedge neg q)wedge (neg pvee q) (5 marks) b) Use a truth table to establish whether the argument below is valid. pwedge qarrow neg r pVneg q

a) <br />(i) To determine whether the statement $(p\wedge q)\vee (\neg p\vee (p\wedge \neg q))$ is a tautology or contradiction, we can construct a truth table:<br /><br />| p | q | p ∧ q | ¬p | p ∧ ¬q | ¬p ∨ (p ∧ ¬q) | (p ∧ q) ∨ (¬p ∨ (p ∧ ¬q)) |<br />|---|---|-------|----|-------|-------------|---------------------|<br />| T | T | T | F | F | F | T |<br />| T | F | F | F | F | F | F |<br />| F | T | F | T | F | F | F |<br />| F | F | F | T | T | T | T |<br /><br />From the truth table, we can see that the statement $(p\wedge q)\vee (\neg p\vee (p\wedge \neg q))$ is a tautology because it is true for all possible combinations of truth values for p and q.<br /><br />(ii) To determine whether the statement $(p\wedge \neg q)\wedge (\neg p\vee q)$ is a tautology or contradiction, we can construct a truth table:<br /><br />| p | q | ¬q | p ∧ ¬q | ¬p | ¬p ∨ q | (p ∧ ¬q) ∧ (¬p ∨ q) |<br />|---|---|----|-------|----|-------|------------------|<br />| T | T | F | F | F | T | F |<br />| T | F | T | T | F | F | F |<br />| F | T | F | F | T | T | T |<br />| F | F | T | F | T | T | T |<br /><br />From the truth table, we can see that the statement $(p\wedge \neg q)\wedge (\neg p\vee q)$ is neither a tautology nor a contradiction because it is true for some combinations of truth values for p and q.<br /><br />b) To determine whether the argument $p\wedge q\rightarrow \neg r$ and $pV\neg q$ is valid, we can construct a truth table:<br /><br />| p | q | r | ¬r | p ∧ q | ¬q | p ∧ q → ¬r | p ∨ ¬q | (p ∧ q → ¬r) ∧ (p ∨ ¬q) |<br />|---|---|---|----|-------|----|-----------|-------|---------------------|<br />| T | T | T | F | T | F | F | T | F |<br />| T | T | F | T | T | F | T | T | T |<br />| T | F | T | F | F | T | F | T | F |<br />| T | F | F | T | F | T | T | T | T |<br />| F | T | T | F | F | F | F | F | F |<br />| F | T | F | T | F | F | T | F | F |<br />| F | F | T | F | F | T | F | T | F |<br />| F | F | F | T | F | T | T | T | T |<br /><br />From the truth table, we can see that the argument $p\wedge q\rightarrow \neg r$ and $pV\neg q$ is valid because the compound statement $(p\wedge q\rightarrow \neg r)\wedge (p\vee \neg q)$ is true for all possible combinations of truth values for p, q, and r.