If $A \subseteq B, $ then $f^{-1}(A) \subseteq f^{-1}(B)$

Question

Let $f: X \xrightarrow{} Y $and $A,B \subseteq Y. $ Show that if $A \subseteq B, $ then $f^{-1}(A) \subseteq f^{-1}(B)$

I'm having a tough time getting started with this one so if someone could push me in the right direction, I'd greatly appreciate it.

Welcome to Mathematics Stack Exchange. You want to show that if $x\in f^{-1}(A) $ then $x\in f^{-1}(B)$. So assume $x\in f^{-1}(A)$. That means $f(x)\in A$. Since $A\subseteq B\dots$ — J. W. Tanner, Apr 26 '20 at 22:46

score 2 · Accepted Answer · answered Apr 26 '20 at 22:44

2

Hint: $$x \in f^{-1}(A) \iff f(x) \in A.$$

answered Apr 26 '20 at 22:44

Sahiba Arora

10,847

score 2 · Answer 2 · edited Apr 28 '20 at 11:42

2

$$x \in f^{-1}(A) \iff f(x) \in A \implies f(x) \in A\subseteq B \implies f(x) \in B \iff x \in f^{-1}(B).$$ Hence $f^{-1}(A) \subseteq f^{-1}(B)$. Generally $f^{-1}$ respects set operation.

edited Apr 28 '20 at 11:42

Mark

649

answered Apr 27 '20 at 13:19

Noob mathematician

2,028

If $A \subseteq B, $ then $f^{-1}(A) \subseteq f^{-1}(B)$

2 Answers2

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