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Given that:

$A\subset B$

How do we know that:

$A\cap B = A$

and are there any other rules related to subsets, that influence the typical intersection, union, and complement operations?

I was calculating $P(B|A)$ and the relationship above tripped me up.

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    Draw a picture. – 2017-01-22
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    Drawing the picture helped. Thanks! – 2017-01-22

1 Answers 1

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Of course every element of the intersection of $A$ and $B$ is an element of $A$. $$A\supseteq A\cap B$$

Now, the meaning of set $A$ being a subset of set $B$, is that every element of $A$ is also an element of $B$.   From that we can infer that every element of $A$ is an element of $A$ and of $B$. $$A\subseteq B ~\implies~ A\subseteq A\cap B$$

Thus we can conclude that the set $A$ equals the intersection of $A$ and $B$. $$\therefore~~A\subseteq B ~\implies~ A = A\cap B$$