- #1

SithsNGiggles

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## Homework Statement

The question is "Prove

[itex]f(\bigcap_{\alpha \in \Omega} A_\alpha) \subseteq \bigcap_{\alpha \in \Omega} f(A_\alpha)[/itex]

where [itex]f:X \rightarrow Y[/itex] and [itex]\{A_\alpha : \alpha \in \Omega\}[/itex] is a collection of subsets of [itex]X[/itex].

Also, prove the statement's equality when [itex]f[/itex] is an injective function.

## Homework Equations

## The Attempt at a Solution

Let [itex]f(x) \in f(\bigcap_{\alpha \in \Omega} A_\alpha)[/itex].

Then [itex]\exists x \in \bigcap_{\alpha \in \Omega} A_\alpha[/itex],

i.e. [itex]\exists x \in A_\alpha \forall \alpha \in \Omega[/itex].

It follows that [itex]\exists f(x) \in f(A_\alpha) \forall \alpha \in \Omega[/itex],

i.e. [itex]f(x) \in \bigcap_{\alpha \in \Omega} f(A_\alpha)[/itex].

Thus [itex]f(x) \in f(\bigcap_{\alpha \in \Omega} A_\alpha) \Rightarrow f(x) \in \bigcap_{\alpha \in \Omega} f(A_\alpha)[/itex], and hence,

[itex]f(\bigcap_{\alpha \in \Omega} A_\alpha) \subseteq \bigcap_{\alpha \in \Omega} f(A_\alpha)[/itex].

I tried following a similar thought process in proving the reverse, but I end up showing that the RHS is a subset of the LHS. Although it works out for the injective f, where does it go wrong when f is not injective?