Write the following logical statement symbolically

Question

I was given "If $f'(x)>0$ for every $x\in(a, b),$ then $f$ is increasing on $[a, b].$" and I need to write it out symbolically. I have gotten $$∀x\in(a,b) ~(f'(x)>0)\to (x<y\to f(x)<f(y))$$

I don't know how to rewrite $f'(x)>0$ in more basic terms like I did with increasing. Can anybody help me with this?

Answer 1

It looks almost correct to me, save that you need to quantify the free variables on the RHS.$$∀x\in(a,b) ~(f'(x)>0)\to \boxed{????}(x<y\to f(x)<f(y))$$

Answer 2

You need to qualify $x$ and $y$ in the 'then' part. Otherwise, you're nearly there. Here's what I would write: $$(x\in(a,b) \implies f'(x)>0)\implies[((x\in[a,b])\land(y\in[a,b])\land (x<y))\implies f(x)\le f(y)].$$