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#### bergausstein

##### Active member

- Jul 30, 2013

- 191

can you show me how to draw the first one with the union of the sets, after that I'll try to illustrate the second statement. just want to get an idea how to go about it.thanks!

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- Thread starter
- #1

- Jul 30, 2013

- 191

can you show me how to draw the first one with the union of the sets, after that I'll try to illustrate the second statement. just want to get an idea how to go about it.thanks!

- Jan 30, 2012

- 2,518

I would draw it like this.

The idea is that red union is done first and then one adds the blue set. In the end we are interested in the colored region, which is the same in both cases.

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- Jul 30, 2013

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- #4

- Jul 30, 2013

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- Aug 7, 2013

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- #6

- Jul 30, 2013

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- Mar 10, 2012

- 834

Good job!this is what i tried for the associativity of intersection.

the black circle is A, the red one is B and the blue one is C.

View attachment 1151

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To me this is okay. But as Evgeny.Makarov pointed out this might not be okay to someone else. Cuz he might say that 'no this does not illustrate the identity correctly' and no one can do anything about it. Don't give it too much importance. Just be sure to understand why $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$ is true. Can you show this without a diagram?

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- #8

- Jul 30, 2013

- 191

$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

$\{1,2,3\}\cup \{6\}=\{1,2,3,4,5,6\}\cap \{1,2,3,4,6,7,8\}$

$\{1,2,3,6\}=\{1,2,3,6\}$

but i know there's a more general way to show why that is true. can you show me your work? thanks! I'm weak when it comes to generalizing.

- Mar 10, 2012

- 834

The general way of showing that $X=Y$ is to show that $X\subseteq Y$ and $Y\subseteq X$.

$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

$\{1,2,3\}\cup \{6\}=\{1,2,3,4,5,6\}\cap \{1,2,3,4,6,7,8\}$

$\{1,2,3,6\}=\{1,2,3,6\}$

but i know there's a more general way to show why that is true. can you show me your work? thanks! I'm weak when it comes to generalizing.

Here $X=A\cup(B\cap C)$ and $Y=(A\cap B)\cap (A\cup C)$. Let $x\in X$. Can you show that $x$ is in $Y$ too?

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- #10

- Jul 30, 2013

- 191

am i right?

and for educational purposes can anybody show me your complete work for proving the statement $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$

thanks!