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) (A B) ( A B) This statement is a tautology. Explanation:* This is De Morgan's Law, which states that (A B) is logically equivalent to ( A B). Since both sides of the biconditional are logically equivalent, the biconditional statement is always true. v) (A B) ( A B) This statement is a contradiction. Explanation:* (A B) ( A B) Using De Morgan's Law, ( A B) (A B): (A B) (A B) Let X = (A B). The expression becomes X X. A statement X conjoined with its negation 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 (P Q). Statement (b) can be represented as ( P) Q. We construct truth tables for both statements: Truth table for (P Q): | P | Q | P Q | (P Q) | |:---:|:---:|:-------------------:|:--------------------------:| | T | T | T | F | | T | F | F | T | | F | T | F | T | | F | F | T | F | Truth table for ( P) Q: | P | Q | P | ( P) Q | |:---:|:---:|:--------:|:--------------------------:| | T | T | F | F | | T | F | F | T | | F | T | T | T | | F | F | T | F | Since the final columns for (P Q) and ( P) 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 B) C Premise 2: C Conclusion: A This argument is an application of Modus Tollens. From Premise 1 and Premise 2, we can validly conclude (A B). (A B) is logically equivalent to A 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 ( 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 B) would be false, making the first premise (A B) C true (since a false antecedent implies anything). If Kunle didn't go to New York ( C is true), then both premises are true, but the conclusion ( 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 Q. a. Converse: Q P If I must remember my PIN, then I must withdraw with my debit card. b. Inverse: P Q If I must not withdraw with my debit card, then I must not remember my PIN. c. Contra-positive: Q P If I must not remember my PIN, then I must not withdraw with my debit card. That's 2 down. 3 left today — send the next one.