Help with logical symbols question

Question

I had a homework question which I wasn't really able to do.

The question is A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is said to be increasing on $\mathbb{R}$ if it satisfies the following property:

If $x < y,$ then $f(x) < f(y).$

(a) Use logical symbols (that is, quantifiers and connectives) to write down the definition of an increasing function.

(b) Write the negation of part (a) without using the negation symbol.

So for part (a) would it be $x < y \implies f(x) < f(y)$ and (b) $x > y$ and not $f(x) > f(y)$

Not sure what the logical symbol for and not is.

any help is appreciated.

Answer 1

$(a):\quad \forall x\in \mathbb R,\forall y\in \mathbb R\Big( (x\lt y)\to (f(x)\lt f(y)\Big)$

$(b): \quad \lnot\Big(\forall x \in \mathbb R, \forall y\in \mathbb R\big((x\lt y)\to (f(x)\lt f(y))\big)\Big).$

which is equivalent to $(b): \quad \exists x \in \mathbb R,\exists y\in \mathbb R \Big(\lnot \big(\lnot(x\lt y) \lor (f(x)\lt f(y))\big)\Big)$

which is equivalent to $(b): \quad \exists x \in \mathbb R, \exists y \in \mathbb R\Big((x \lt y)\land (f(x)\geq f(y))\Big)$

Answer 2

You're right for A, but not for B. When we negate $P \implies Q$ you must remember the disjunctive form - that is, recall that

$$ (P \implies Q) \iff (\sim P \ \vee Q) $$

(that is, $P$ implies $Q$ if and only if (or can be read "is equivalent to") not $P$ or $Q$).

So to negate an implication, we must use the latter form and use de Morgan's law - the negation is then $P \ \wedge \sim Q$. So your desired negation is "A function $f$ is not increasing if $x < y$ and $f(x) \geq f(y)$.