Does function: $ f: \mathbb{Z} \to \mathbb{Z} $ exist for these statements

Question

Does function: $ f: \mathbb{Z} \to \mathbb{Z} $ exist, such that this statement is true: $$(\forall{x} \in \mathbb{Z}:f(x) \geq 2)\Rightarrow(\forall{x}\in \mathbb{Z}:f(x)<10)$$ and this statement is false: $$ \forall{x} \in \mathbb{Z}:(f(x) \geq 2\Rightarrow f(x)<10) $$ Now I understand that the second statement is stronger. But I am not sure how I would go about proving whether the function exists or not on paper.

Any help would be greatly appreciated.

Answer 1

Recall the following truth table for the conditional $p\Rightarrow q$:

p q | ( p => q )

T T | T
T F | F
F T | T
F F | T

You see that you can make an implication of the form $p\Rightarrow q$ true by choosing $p$ to be false (and $q$ to have any truth value). To make the claim $$(\forall{x} \in \mathbb{Z}:f(x) \geq 2)\Rightarrow(\forall{x}\in \mathbb{Z}:f(x)<10)$$ true, I can thus choose a function $f$, such that $\exists x\in\Bbb{Z}\colon f(x)<2$ is a true statement, i.e. the function should take a value less than $2$ at at least one integer point. You also want $$\forall{x} \in \mathbb{Z}:(f(x) \geq 2\Rightarrow f(x)<10)$$ to be false. I can make the conditional $p\Rightarrow q$ false by choosing $p$ to be true and $q$ to be false, so I need at least one integer point $x$, such that $f(x)\geq 2$ is true, but $f(x)<10$ is false (i.e. $f(x)\geq 10$ is true). This can be done by simply choosing a point $x$, such that $f(x)\geq 10$.

Summa summarum, I need a point $x_1$, such that $f(x_1)<2$, and another point $x_2$, such that $f(x_2)\geq 10$. Any function with these properties works.

Answer 2

In order for the second statement to be false there has to be some $x_0$ such that $f(x_0)\ge 2$ and yet $\neg(f(x_0)<10)$. That is, $f(x_0) \ge 10$.

This means that the conclusion of the first statement will be false, so the only way to make the entire implication true is to make sure that the hypothesis is false too. Therefore there must be an $x_1$ such that $f(x_1)<2$.

Thus we see that your requirements will be satisfied as long as there is at least one function value that is $\ge 10$ and one function value that is $<2$.

You'll have considerable freedom in choosing such a function.