# Thread: Set equality

1. ## Set equality

I need to prove this plz help

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A,B,C \ and\ D\ are\ non\ empty\ sets\ with\ \left|A\right|=\left|B\right| and\ \left|C\right|=\left|D\right|$

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1. Prove\ that\ \left|A\times C\right|=\left|B \times D\right|$

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2. \left|A^{n}\right|= \left|B^{n}\right|
$

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3.\left|P\left(A\right)\right|=\left|P\left(B\righ t)\right|

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With P i mean the power set

thx guys

2. $|A|=|B|\ \mbox{and}\ |C|=|D|\ \Rightarrow\ \exists\ \mbox{bijections}\ \mathrm{f}:A\to B\ \mbox{and}\ \mathrm{g}:C\to D$. Now show that the following functions are bijections.

(1) $\mathrm{h}:A\times C\to B\times D,\ \mathrm{h}(a,c)=(\mathrm{f}(a),\mathrm{g}(c))$

(2) $\mathrm{h}:A^n\to B^n,\ \mathrm{h}(a_1,\dots,a_n)=(\mathrm{f}(a_1),\ldots\ ,\mathrm{f}(a_n))$

(2) $\mathrm{h}:\mathcal{P}(A)\to\mathcal{P}(B),\ \mathrm{h}(X)=\{\mathrm{f}(a):a\in X\}$

3. From the given, there are bijections: $f:A \leftrightarrow B\,\& \,g:C \leftrightarrow D$.
Now define a function $\Phi :\left( {A \times C} \right) \leftrightarrow \left( {B \times D} \right),\,\,\Phi (a,c) = \left( {f(a),g(c)} \right)$.
Prove that $\Phi$ is a bijection.

I have no idea what the notation in part b means.

For part c, use the $\chi$ function: $\chi _E (x) = \left\{ {\begin{array}{lr} 1 & {x \in E} \\ 0 & {x \notin {\rm E}} \\ \end{array}} \right.$

4. Originally Posted by kuntah
I need to prove this plz help

$
A,B,C \ and\ D\ are\ non\ empty\ sets\ with\ \left|A\right|=\left|B\right| and\ \left|C\right|=\left|D\right|$

$
1. Prove\ that\ \left|A\times C\right|=\left|B \times D\right|$
Hint: $|A \times C| = |A||C|$

2. $\left|A^{n}\right|= \left|B^{n}\right|
$
Hint: $\left| A^n \right| = | \underbrace{A \times A \times A \cdots A}_{\mbox{n times}}| = \underbrace {|A||A||A| \cdots |A|}_{\mbox{n times}}$

3. $\left|P\left(A\right)\right|=\left|P\left(B\right) \right|

$

With P i mean the power set
Hint: $|P(A)| = 2^{|A|}$

EDIT: ...ok, so both JaneBennet and Plato have different methods from mine, and theirs are similar, so chances are what I did was foolishness . you may ignore this post. i'll leave it up for anyone to comment on it.

5. Originally Posted by Jhevon
EDIT: ...ok, so both JaneBennet and Plato have different methods from mine, and theirs are similar, so chances are what I did was foolishness . you may ignore this post. i'll leave it up for anyone to comment on it.
The posters proved this for any sets in general. You assumed it were finite sets and applied finite numbers.

6. Originally Posted by ThePerfectHacker
The posters proved this for any sets in general. You assumed it were finite sets and applied finite numbers.
i thought so, thanks