Suppose that the universe of x and y consists of the numbers 2 and 5. Write a statement logically equivalent to the given statement, and which has no quantifiers. Determine the truth value of your statement.
i) $\forall x, 10x \leq 50$
ii) $\exists y, (2y \lt 6) \rightarrow (y = 3)$
iii) $\exists x, \forall y, xy \geq 10$
For i, can I assign $P(x) = 10x \leq 50$ and say $\forall x, 10x \leq 50 \Leftrightarrow P(2) \land P(5)$?
If I can, then could I just do the same for the other two questions?
On
For i, can I assign $P(x) = 10x \leq 50$ and say $\forall x, 10x \leq 50 \Leftrightarrow P(2) \land P(5)$?
You can't use the $\Leftrightarrow$ here, as that is about all possible domains. So you merely write down what the quantified statement amounts to relative to this specific domain, which is $10 \cdot 2 \le 50 \land 10 \cdot 5 \le 50$
You could introduce new predicates, but it obscures the evaluation. Why not just write the following, which make the result obvious? In the domain of discourse $\{2,5\}$ then:
$$\forall x~(10x\leq 50) ~\iff~ (10\cdot 2\leq 50)\wedge (10\cdot 5\leq 50)$$