Discrete Mathematics

[[1,1],[1,1]]

Step 1: The join (∨) of Boolean matrices uses the rule: cᵢⱼ = 1 if either aᵢⱼ = 1 or bᵢⱼ = 1, else 0. Step 2: Position (1,1): 0 ∨ 1 = 1 (since b₁₁=1). Step 3: Position (1,2): 1 ∨ 0 = 1 (since a₁₂=1). Position (2,1): 1 ∨ 0 = 1. Position (2,2): 0 ∨ 1 = 1. Step 4: Combine all positions: A ∨ B = [[1,1],[1,1]]. Final: The join operation gives all 1s here because every position has at least one 1 in A or B. Think of ∨ as the logical OR — 1 if any input is 1.

* is binary on ℤ because a + 3ab - 5b² ∈ ℤ for all a, b ∈ ℤ

Step 1: For * to be binary on ℤ, we need a * b = a + 3ab - 5b² ∈ ℤ for all a, b ∈ ℤ. Step 2: Since a, b ∈ ℤ and ℤ is closed under multiplication, ab ∈ ℤ and b² ∈ ℤ. Step 3: Since ℤ is closed under addition, 3ab = ab + ab + ab ∈ ℤ and 5b² ∈ ℤ. Step 4: Therefore a + 3ab - 5b² is a sum/difference of integers, which is always an integer. Final: The operation * is binary on ℤ. The key is verifying closure — every combination of a, b ∈ ℤ must yield a result in ℤ.

When p is T and q is F

Step 1: The conditional p → q means 'If p, then q'. We analyze all four combinations of truth values. Step 2: p=T, q=T: The hypothesis is true and so is the conclusion → T. Step 3: p=T, q=F: The hypothesis is true but the conclusion is false — this violates the implication → F. Step 4: p=F, q=T or p=F, q=F: When the hypothesis is false, the conditional is vacuously true → T in both cases. Final: p → q is FALSE only when p is TRUE and q is FALSE. Think of it as a promise: if you make a promise (p=T) and break it (q=F), the statement is false.

If a number is not divisible by 2, then it is not divisible by 6

Step 1: Identify the original statement p → q. Here p: 'a number is divisible by 6', q: 'it is divisible by 2'. Step 2: The contrapositive is formed by negating both p and q and swapping them: ¬q → ¬p. Step 3: ¬q: 'a number is NOT divisible by 2', ¬p: 'it is NOT divisible by 6'. Step 4: Contrapositive: 'If a number is not divisible by 2, then it is not divisible by 6'. Final: The contrapositive ¬q → ¬p is logically equivalent to p → q. Note: the converse is q → p (different statement) and inverse is ¬p → ¬q (also different).