Verify whether the following statements are tautologies or contradictions and show the equivalence of compound statements using truth tables: iv) (A B) ( A B), v) (A B) ( A B), and compare a) 'It is n

Here are the solutions to the remaining questions:

2) Verify whether each of the following statements is a tautology or contradiction:

• iv) $\sim (A \lor B) \Leftrightarrow (\sim A \land \sim B)$ This statement is a tautology.

  • Explanation: This is De Morgan's Law, which states that $\sim (A \lor B)$ is logically equivalent to $(\sim A \land \sim B)$. Since both sides of the biconditional are logically equivalent, the biconditional statement is always true.

• v) $(A \lor B) \land (\sim A \land \sim B)$ This statement is a contradiction.

  • Explanation: $$(A \lor B) \land (\sim A \land \sim B)$$ Using De Morgan's Law, $(\sim A \land \sim B) \equiv \sim (A \lor B)$: $$(A \lor B) \land \sim (A \lor B)$$ Let $X = (A \lor B)$. The expression becomes $X \land \sim X$. A statement $X$ conjoined with its negation $\sim X$ is always false. Therefore, the statement is always false.

3) Consider these compound statements;

• a) It is not true that the sky is dark, if and only if it will rain
• b) The sky is not dark if and only if it will rain Using truth tables, show that these compound statements are equivalent.

Let $P$: "The sky is dark"
Let $Q$: "It will rain"

Statement (a) can be represented as $\sim (P \Leftrightarrow Q)$.
Statement (b) can be represented as $(\sim P) \Leftrightarrow Q$.

We construct truth tables for both statements:

Truth table for $\sim (P \Leftrightarrow Q)$:

$P$	$Q$	$P \Leftrightarrow Q$	$\sim (P \Leftrightarrow Q)$
T	T	T	F
T	F	F	T
F	T	F	T
F	F	T	F

Truth table for $(\sim P) \Leftrightarrow Q$:

$P$	$Q$	$\sim P$	$(\sim P) \Leftrightarrow Q$
T	T	F	F
T	F	F	T
F	T	T	T
F	F	T	F

Since the final columns for $\sim (P \Leftrightarrow Q)$ and $(\sim P) \Leftrightarrow Q$ are identical, the two compound statements are equivalent.

4) Determine the validity of the following argument regarding a music concert:

• If Kunle passes the audition and wins first prize, then he will get a trip to New York.
• Eventually, Kunle didn't go to New York.
• Hence, he failed the audition.

Let $A$: "Kunle passes the audition"
Let $B$: "Kunle wins first prize"
Let $C$: "Kunle will get a trip to New York"

The argument can be symbolized as:
Premise 1: $(A \land B) \Rightarrow C$
Premise 2: $\sim C$
Conclusion: $\sim A$

This argument is an application of Modus Tollens. From Premise 1 and Premise 2, we can validly conclude $\sim (A \land B)$.
$\sim (A \land B)$ is logically equivalent to $\sim A \lor \sim B$ (by De Morgan's Law).
This means that either Kunle failed the audition OR Kunle did not win first prize (or both).
The conclusion states that Kunle failed the audition ($\sim A$). This is not necessarily true. It is possible that Kunle passed the audition ($A$ is true) but did not win first prize ($B$ is false). In this scenario, $(A \land B)$ would be false, making the first premise $(A \land B) \Rightarrow C$ true (since a false antecedent implies anything). If Kunle didn't go to New York ($\sim C$ is true), then both premises are true, but the conclusion ($\sim A$) would be false.

Therefore, the argument is invalid.

5) Consider the conditional statement;
"If I must withdraw with my debit card, then I must remember my PIN"
Hence, generate the:

Let $P$: "I must withdraw with my debit card"
Let $Q$: "I must remember my PIN"
The original conditional statement is $P \Rightarrow Q$.

• a. Converse: $Q \Rightarrow P$

  • If I must remember my PIN, then I must withdraw with my debit card.

• b. Inverse: $\sim P \Rightarrow \sim Q$

  • If I must not withdraw with my debit card, then I must not remember my PIN.

• c. Contra-positive: $\sim Q \Rightarrow \sim P$

  • If I must not remember my PIN, then I must not withdraw with my debit card.