Question

Let A be a nonempty set and let P(x) and Q(x) be open statements. Consider the two statements (i) ∀x ∈ A, [P(x)∨Q(x)] and (ii) [∀x ∈ A, P(x)]∨[∀x ∈ A, Q(x)]. Argue whether (i) and (ii) are (logically) equivalent or not. (Can you explain your answer mathematically and by giving examples in plain language ? In the latter, for example, A = {all the CU students}, P(x) : x has last name starting with a, b, ..., or h, and Q(x) : x has last name starting with i, j, ..., or z.) If they are, why ? If not, can you say at least which of the two always implies the other ? (Can you explain your answer mathematically and by giving examples in plain language ?)

Answer 1