Proofs. Below, in the first three questions, A and B denote sets. (a)Prove or disprove, using any method, the following statement: |P(A − B)| = 2|A|−|B|, where P(X) is the power-set of X.rove that A − (B ∩ A) = A − B using the set identities as well as the set subtraction law: A − B = A ∩ B. (c)Prove that A − (B ∩ A) = A − B, but this time using definitions only (of sets, set operations, set relations, etc.), . You should NOT use set identities. (d)Prove by induction that for any positive integer n, 4 divides 32n − 1. (e)Is the statement “If √13 ̸ ∈ Q, then √25 ̸ ∈ Q” True or False? Explain your answer in detail (proof something if necessary). You can use the fact that if n is an integer such that 13 divides n2, then 13 divides n. (f) Given the following clauses: (¬P ∨ W ) ∧ (P ∨ Q ∨ T ) ∧ (¬S) ∧ (P ∨ Q ∨ S) ∧ (¬Q ∨ T ) ∧ (¬T ) ∧ (¬W ) Perform the smallest possible resolution refutation, that is, prove the above CNF formula is unsatisfiable (i.e., a contradiction) in the smallest number of steps. (g) Prove by resolution that KB |= φ, where: KB = (¬P ∨ W ) ∧ (P ∨ Q ∨ T ) ∧ (¬S) ∧ (P ∨ Q ∨ S) ∧ (¬T ) ∧ (¬W ) and φ = Q ∧ ¬T Provide a proof showing all the steps of your reasoning.

d). To prove that for any positive integer n, 4 divides 3^(2n) - 1, we will use mathematical induction. Step 1: Base Case Let's check the base case, n = 1: 3^(2(1)) - 1 = 3^2 - 1 = 9 - 1 = 8 Since 8 is divisible by 4 (8 ÷ 4 = 2), the base case holds. Step 2: Inductive Hypothesis Assume that for some positive integer k, 4 divides 3^(2k) - 1. This is our inductive hypothesis. Step 3: Inductive Step We need to prove that if the statement is true for k, then it is also true for k + 1. Consider the expression for k + 1: 3^(2(k + 1)) - 1 = 3^(2k + 2) - 1 We can rewrite 3^(2k + 2) as (3^(2k) * 3^2): 3^(2k + 2) - 1 = (3^(2k) * 3^2) - 1 = (3^(2k) * 9) - 1 We can further rewrite this as: (3^(2k) * 9) - 1 = 9 * (3^(2k)) - 1 = (8 + 1) * (3^(2k)) - 1 = 8 * (3^(2k)) + (3^(2k)) - 1 Now, let's consider the term 8 * (3^(2k)). According to our inductive hypothesis, 4 divides 3^(2k) - 1, so we can express it as: 3^(2k) - 1 = 4m, where m is an integer. Substituting this into our expression: 8 * (3^(2k)) + (3^(2k)) - 1 = 8 * (4m + 1) + (3^(2k)) - 1 = 32m + 8 + (3^(2k)) - 1 = 32m + (3^(2k)) + 7 Notice that 32m is divisible by 4 since it is a multiple of 4, and 7 is not divisible by 4. Therefore, we have expressed (3^(2(k + 1))) - 1 as the sum of a multiple of 4 and a number that is not divisible by 4. Thus, we have shown that if the statement is true for k, then it is also true for k + 1. Step 4: Conclusion Since the base case holds, and we have established the inductive step, we can conclude by mathematical induction that for any positive integer n, 4 divides 3^(2n) - 1.