## cinar Group Title $Let f:A \rightarrow B  be a given function. Prove that f is one-to-one (injective)  \Leftrightarrow f(C\cap D)=f(C)\cap f(D)  for every pair of sets C and D in A$ one year ago one year ago

1. satellite73 Group Title

Let $f:A\rightarrow B$ be a given function. Prove that f is one-to-one (injective) $\leftrightarrow f(C\cap D)=f(C)\cap f(D)$ for every pair of sets C and D in A

2. cinar Group Title

$Let f:A\rightarrow B be a given function. Prove that f is one-to-one (injective) \\\Leftrightarrow f(C\cap D)=f(C)\cap f(D) for every pair of sets C and D in A$

3. satellite73 Group Title

i was just rewriting so i could read it, i am not sure i know how to do it

4. satellite73 Group Title

well one way is trivial, since $$f(A\cap B)\subset f(A)\cap f(B)$$ for any $$f$$

5. satellite73 Group Title

or does that need clarification as well? we can write it out if you like

6. cinar Group Title

yes we need to right it..

7. satellite73 Group Title

ok

8. cinar Group Title

why letter is not separated

9. satellite73 Group Title

suppose $$z\in f(A\cap B)$$ then $$z=f(x)$$ for some $$x\in A\cap B$$ making $$x\in A$$ and $$x\in B$$ so $$z\in f(A)$$ and $$z\in f(B)$$ therefore $$z\in f(A)\cap f(B)$$

10. satellite73 Group Title

this shows for any $$f$$ you have $$f(A\cap B)\subset f(A)\cap f(B)$$

11. satellite73 Group Title

now we need to prove that if $$f$$ in injective, we have $$f(A\cap B)=f(A)\cap f(B)$$ since we already have containment one way, this amounts to showing $f(A)\cap f(B)\subset f(A\cap B)$

12. satellite73 Group Title

pick a $$z\in f(A)\cap f(B)$$ so there exists a $$x_1$$ in $$A$$ with $$f(x_1)=z$$ and likewise there is a $$x_2$$ in $$B$$ with $$z=x_2$$ now comes the "injective" part since $$f$$ is injective, and $$f(x_1)=f(x_2)=z$$ we know $$x_1=x_2$$ and so $z\in f(A\cap B)$

13. satellite73 Group Title

typo there, i meant "likewise there exists $$x_2\in B$$ with $$f(x_2)=z$$ sorry

14. satellite73 Group Title

so that is the proof one way, that "if $$f$$ is injective, then $$f(A\cap B)=f(A)\cap f(B)$$

15. satellite73 Group Title

other way is easier, since a singleton is a set

16. cinar Group Title

A={x} B={y} you mean like this one..