Question

Consider p(x) and q(x), where x ∈ U = {1, 2}. If the following is true, give a rigorous argument. If it is false, give a counterexample. (Note that “p implies q” is the same as “if p, then q” and also as “p → q.”) (i) (∀x ∈ U, p(x) → q(x)) implies [ (∀x ∈ U, p(x)) → (∀x ∈ U, q(x)) ] ? What about its converse ? (ii) (∃x ∈ U, p(x) → q(x)) implies [ (∃x ∈ U, p(x)) → (∃x ∈ U, q(x)) ] ? What about its converse ?

Answer 1