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Am trying to figure out if all three of these expressions:

1) $B \subseteq A$
2) $B \subset A$
3) $B = A$

imply that 4) $P(A \cap B) = P(B)$ ?

My thinking is as follows:

1) follows from here
2) is just a more restricted condition than 1), so the implication should hold here as well.
3) because if B = A, then P(A) = P(B) = $P(A \cap B)$

I can't see why 1-3 would not imply 4) from my by Venn diagram sketches except if I'm not handling the case where A is the empty set. If I'm missing something, could someone set me straight?

Michael Szczepaniak
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1 Answers1

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If $B\subseteq A$, then $A\cap B = B$. The same applies to the other two conditions. Thus, if any of them is satisfied, then you have $\textbf{P}(A\cap B) = \textbf{P}(B)$. In order to prove it, you may consider the following reasoning. Clearly, $A\cap B\subseteq B$. Based on the assumption of $B\subseteq A$, one has \begin{align*} x\in B \Longrightarrow x\in A\Longrightarrow x\in A\cap B \Longrightarrow (B\subseteq A\cap B). \end{align*} from whence it results that $A\cap B = B$.

user0102
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