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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 $(A' \cap B') \cup A=A \cup B'$?2012-03-25
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    Does $X^\prime$ denote the complement of $X$?2012-03-25
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    yes sir Kannappan Sampath.2012-03-25
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    @AndreaMori, I'm sorry I didn't get you. I don't have any X in my question.2012-03-25
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    OK. What are $A'$ and $B'$? @JCD2012-03-25
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    Complement of set A and Complement of set B2012-03-25
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    When you say complement of set $A$, complement of set from what? I mean, when you define complement, you say, all those elements not in $A$, but in a set, that is frequently called the universal set? What is that universal set?2012-03-25
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    @JCD, I was asking about the meaning of the notation: what does it mean when you go from $X$ to $X^\prime$ where $X$ can be just anything.2012-03-25

2 Answers 2

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One way to see this is using a Venn diagram

enter image description here

$A$ 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!2012-03-25
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    @Kannappan Sampath: That is an issue for the question2012-03-25
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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?2012-03-25
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    @KannappanSampath: "AB" stands for "allzeit bereit", it's the german Scout motto2012-03-25
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    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! : )2012-03-25
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    @KannappanSampath Ok, I'll stick to this now. Do I have to edit all my posts to remove my tagline?2012-03-25
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    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,2012-03-25
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    @KannappanSampath Now it doesn't. It's ok, I just didn't know. Of course I'll do so from now on, but I hope it's ok if I don't change all posts I already made.2012-03-25
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    Yes, sure. You do not have to. And, another thing is you will bump a lot of old posts! This floods the main page with old questions pushing all the new ones below. So, never mind about your old posts.2012-03-25