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Let $A, B$ be finite sets. Show that $A \cup B = (A$ \ $B ) \cup (A \cap B) \cup (B$ \ $A)$.

Deduce that $|A| + |B| = |A \cup B| + |A \cap B|$

These are obvious when considering Venn diagrams but I'm at a lose on even how to start to show this algebraically.

I've tried re-writing $(A$ \ $B ) \cup (A \cap B) \cup (B$ \ $A)$ as $(A \cap \neg B) \cup (A \cap B) \cup (B \cap \neg A)$ but I don't know where to go from there.

Any help would be appreciated. Thank you.

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  • $\begingroup$ What is $\neg B$ for a set? $\endgroup$ – martini Jul 29 '14 at 8:24
  • $\begingroup$ @martini I meant everything that's not inside the set B. $\endgroup$ – Elise Jul 29 '14 at 8:34
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Use distributivity laws. For example:

$$(A\cap \neg B)\cup(A\cap B) = (A\cup A) \cap (A\cup B)\cap(A\cup \neg B) \cap (B\cup \neg B) =\\ =A\cap (A\cup B) \cap (A\cup \neg B)\cap U$$ Now, since $U$ is the universal set, $U\cap X=X$ for any $X$. Also, since $A\subseteq A\cup B$, you know that $A\cap (A\cup B)=A$, so you get $$A\cap(A\cup \neg B)$$ and again, vecause $A\subseteq A\cup\neg B$, you have $A\cap(A\cup\neg B)=A$, so in conclusion:

$$(A\cap \neg B)\cup (A\cap B) = A.$$

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  • $\begingroup$ Thank you. Would I be correct in carrying it on to say: $A \cup (B \cap \neg A) = (A \cup B) \cap (A \cup \neg A) = (a \cup B) \cap U = A \cup B$ $\endgroup$ – Elise Jul 29 '14 at 9:06
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    $\begingroup$ @Elise Very good! you are finished! $\endgroup$ – 5xum Jul 29 '14 at 9:08
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On one hand $$ A\setminus B, A \cap B \subseteq A \subseteq A \cup B, \qquad B \setminus A \subseteq B \subseteq A \cup B $$ hence "$\supseteq$" holds.

Now let $x \in A \cup B$. Then $x \in A$ or $x \in B$. If $x \in A$, then either $x \in B$ and hence $x \in A \cap B$ or $x \not\in B$ hence $x \in A \setminus B$. If $x \not\in A$, then as $x \in A \cup B$ we must have $x \in B$ and hence $x \in B \setminus A$. So in all cases $x \in (B \setminus A) \cup (A \cap B) \cup (A \setminus B)$. This proves "$\subseteq$".

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  • $\begingroup$ Thank you. I understand your second paragraph - which is a really nice way of doing it - but I don't understand what you mean by "hence "$\supseteq$" holds"? $\endgroup$ – Elise Jul 29 '14 at 9:11
  • $\begingroup$ As we know (as I wrote) $A -B, A \cap B, B-A \subseteq A \cup B$, we have $$ A \cup B \supseteq (A-B)\cup (A\cap B) \cup (B-A) $$ which is one inclusion of the two we need for the equality you stated (namely "$\supseteq$"). $\endgroup$ – martini Jul 29 '14 at 10:52
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you can't whisk a proof out of thin air! so what assumption are you starting from? the most reasonable assumption is that if two sets $X$ and $Y$ are mutually disjoint then $$ |X \cup Y | = |X| + |Y| \tag{1} $$ see if you can solve the problem using this assumption.

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