I am having trouble understanding the statement that says If $x \in B \cap C$, then $x \in B$ or $x \in C$. Shouldn't this read: If $x \in B \cap C$, then $x \in B$ and $x \in C$? enter image description here

  • $\begingroup$ @StVincent Your modification makes the question impossible to understand. Please be more careful. $\endgroup$ – Did Sep 22 '13 at 9:01
  • $\begingroup$ It is just a typo in the reference. The "or" at the end of the first line is supposed to be an "and". The rest of the argument shows that this is what the author had in mind. $\endgroup$ – Andrés E. Caicedo Sep 22 '13 at 19:03

Yes, you're correct. The definition of set intersection requires that $x \in B \cap C$ if and only if $x \in B$ and $x \in C$.

The given proof is extremely difficult to read, and isn't written in anything even close to standard notation. A proof of the result would read something like

Suppose that $x \in A \cup (B \cap C)$. If $x \in A$, then $x \in A \cup B$ and $x \in A \cup C$ by definition of union, so $x \in (A \cup B) \cap (A \cup C)$. If $x \in B \cap C$, then $x \in B$ and $x \in C$, so by a similar argument, we find that $x$ lies in the correct set.

If $x \in (A \cup B) \cap (A \cup C)$, then $x \in A \cup B$ and $x \in A \cup C$. If $x \in A$, we're done; if $x \notin A$, then $x \in B$ and $x \in C$, so that $x \in B \cap C$ and we're again done.

  • $\begingroup$ @AsafKaragila I was reading the question as "isn't the proof I read in a book wrong," and the answer is affirmative. I didn't mean to imply that the quoted text was correct. $\endgroup$ – user61527 Sep 21 '13 at 22:34
  • $\begingroup$ Oh, right. I misread the question... :-) $\endgroup$ – Asaf Karagila Sep 21 '13 at 22:35
  • $\begingroup$ the second part of the proof was left up to the student. $\endgroup$ – Al Jebr Sep 21 '13 at 22:50

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