Question

Prove that if f and g are functions from A to B and $S_f=S_g$, the $f\left(x\right)=g\left(x\right)\forall x\in A$.

Short answer

Let $x\in A$. Thenrole="math" localid="1668595804992" $S_f\left\{\left(x\right)\right\}=\left\{f\left(y\right)|y\in x=f\left(x\right)\right\}$.

By the same reasoning,$S_g\left\{\left(x\right)=g\left(x\right)\right\}$ .

Because$S_f=S_g$ , we can conclude that $\left\{f\left(x\right)=g\left(x\right)\right\}$, and so necessarily$f\left(x\right)=g\left(x\right)$

Step-by-step solution

Step: 1

Let.$x\in A$ Then$S_f\left(\left\{x\right\}\right)=\left\{f\left(y\right)|y\in x=f\left(x\right)\right\}$.

By the same reasoning,$S_g\left\{\left(x\right)\right\}=\left\{g\left(x\right)\right\}$.

Because $S_f=S_g$, we can conclude that $\left\{f\left(x\right)\right\}=\left\{g\left(x\right)\right\}$, and so necessarily$f\left(x\right)=g\left(x\right)$