What Determines When Two Mathematical Functions Are Equal?

[evinda]

Hello! (Wave)

I am looking at the proof of the following sentence:

Sentence:

Let $f,g$ functions. Then:

$$f=g \leftrightarrow dom(f)=dom(g) \wedge (\forall x \in dom(f)) (f(x)=g(x))$$

Proof:

$$\Rightarrow$$
If $f=g$, then $f \subset g$ and $g \subset f \Rightarrow dom(f) \subset dom(g) \text{ and } dom(g) \subset dom(f)$

So, $dom(f)=dom(g)$.

Also, from the sentence: If $f,g$ functions and $f \subset g$, we have that $\forall x \in dom(f): f(x)=g(x)$
we have that: $\forall x \in dom(f) f(x)=g(x)$

$$\Leftarrow$$

We want to show $f=g$, knowing that $dom(f)=dom(g)$ and $\forall x \in dom(f) f(x)=g(x)$

It suffices to show that $f \subset g$ and $g \subset f$.

Let $t \in f$. Then, $t=<x,f(x)>$, for a $x \in dom(f)$.
But, $dom(f)=dom(g)$ and so:
$$x \in dom(g) \leftrightarrow <x,g(x)> \in g$$
From the hypothesis, $f(x)=g(x)$ and so $t=<x,f(x)> \in g$.

Therefore, $f \subset g$.

In the same way, we show that $g \subset f$.
Therefore, $f=g$.

Could you explain me the $\Leftarrow$ part of the proof? (Thinking)

 

[Evgeny.Makarov]

You'll have to ask a more specific question.