Final answer:
The statement ∃x ∀y(x ≤ y) is true because 1 is less than or equal to every positive integer. The statement ∃x ∃y(x < y) is true because for any positive integer x, there exists another integer y that is greater than x.
Step-by-step explanation:
The question pertains to predicate logic within the realm of mathematics and concerns the evaluation of truth values for given logical statements where the universe of discourse is the set of all positive integers, ℤ⁺.
i. ∃x ∀y(x ≤ y): This statement is asserting the existence of an x for which every y is greater than or equal to x. In the set of positive integers, 1 satisfies this condition, so the statement is true.
ii. ∃x ∃y(x < y): This statement claims there exist some x and y within the positive integers where x is less than y. This is true as for any positive integer, one can find another higher integer (e.g., x=1 and y=2).