# Proving that two set are equal: $(A'\cap B')\cup A=A\cup B'.$

I need help understanding how this solution was made:

(A'∩B')∪A=A∪B'
[(A')'∪B]∪A=A∪B'
(A∪B')∪A=A∪B'
A∪(A∪B')=A∪B'
A∪B'=A∪B'

I don't really know how our instructor arrived to that answer. How to prove that (A'∩B')∪A=A∪B'?

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It is not clear what is the question: Is this question asking us to help you with how to prove that $(A' \cap B') \cup A=A \cup B'$? –  user21436 Mar 25 '12 at 9:23
Does $X^\prime$ denote the complement of $X$? –  Andrea Mori Mar 25 '12 at 9:24
yes sir Kannappan Sampath. –  Zhianc Mar 25 '12 at 9:24
@AndreaMori, I'm sorry I didn't get you. I don't have any X in my question. –  Zhianc Mar 25 '12 at 9:26
OK. What are $A'$ and $B'$? @JCD –  user21436 Mar 25 '12 at 9:27

One way to see this is using a Venn diagram

$A$ is the blue and green

$B$ is the blue and white

$A'$ is the yellow and white

$B'$ is the yellow and green

So $A' \cap B'$ is the yellow, and $(A' \cap B') \cup A$ is the yellow, blue and green

while $A \cup B'$ is also the blue, green and yellow, so they are equal.

Another approach is $$A \cup B' = A \cup [(A \cap B') \cup (A' \cap B')] = [A \cup (A \cap B')] \cup (A' \cap B')$$ $$= A \cup (A' \cap B') = (A' \cap B') \cup A$$

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It may be useful mention what $A'$ here means! –  user21436 Mar 25 '12 at 9:49
@Kannappan Sampath: That is an issue for the question –  Henry Mar 25 '12 at 9:54

supposing that $A'$ and $B'$ are the complements of $A$ and $B$ with respect to some set $X$, we have for $x \in X$:

If $x \in (A' \cap B') \cup A$ then if $x \in A$ obviously $x \in A \cup B'$. If $x \not\in A$, we must have $x \in A' \cap B'$, so $x \in B'$ and therefore $x \in A \cup B'$.

If $x \in A \cup B'$: If $x \in A$ then $x \in (A' \cap B') \cup A$, otherwise i. e. if $x \not\in A$ we must have $x \in B'$. As $x \not\in A$ we have $x \in A'$ and therefore $x \in A' \cap B' \subseteq (A' \cap B') \cup A$.

Now we have shown $(A' \cap B') \cup A \subseteq B' \cup A$ and $B' \cup A \subseteq (A' \cap B') \cup A$. So the two sets are equal.

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I assume HTH stands for "Hope that helps". But what is AB? –  user21436 Mar 25 '12 at 9:47
@KannappanSampath: "AB" stands for "allzeit bereit", it's the german Scout motto –  martini Mar 25 '12 at 9:50
I am sorry to tell you that faq on the site explicitly forbids you from signing your posts with any tag line. So, I'd request you to not sign your posts. "Hope that helps" looks OK to me but then, it might be good to write it out fully! : ) –  user21436 Mar 25 '12 at 9:54
@KannappanSampath Ok, I'll stick to this now. Do I have to edit all my posts to remove my tagline? –  martini Mar 25 '12 at 9:56
I am not sure, if that is required. But, yes, in the future, it would be the good thing to do. Hope this does not come across to you wrongly. Regards, –  user21436 Mar 25 '12 at 9:58