# Set equality

• Jan 24th 2008, 02:51 PM
kuntah
Set equality
I need to prove this plz help

$\displaystyle A,B,C \ and\ D\ are\ non\ empty\ sets\ with\ \left|A\right|=\left|B\right| and\ \left|C\right|=\left|D\right|$
$\displaystyle 1. Prove\ that\ \left|A\times C\right|=\left|B \times D\right|$
$\displaystyle 2. \left|A^{n}\right|= \left|B^{n}\right|$
$\displaystyle 3.\left|P\left(A\right)\right|=\left|P\left(B\righ t)\right|$
With P i mean the power set

thx guys
• Jan 24th 2008, 03:12 PM
JaneBennet
$\displaystyle |A|=|B|\ \mbox{and}\ |C|=|D|\ \Rightarrow\ \exists\ \mbox{bijections}\ \mathrm{f}:A\to B\ \mbox{and}\ \mathrm{g}:C\to D$. Now show that the following functions are bijections.

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

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

(2) $\displaystyle \mathrm{h}:\mathcal{P}(A)\to\mathcal{P}(B),\ \mathrm{h}(X)=\{\mathrm{f}(a):a\in X\}$
• Jan 24th 2008, 03:15 PM
Plato
From the given, there are bijections: $\displaystyle f:A \leftrightarrow B\,\& \,g:C \leftrightarrow D$.
Now define a function $\displaystyle \Phi :\left( {A \times C} \right) \leftrightarrow \left( {B \times D} \right),\,\,\Phi (a,c) = \left( {f(a),g(c)} \right)$.
Prove that $\displaystyle \Phi$ is a bijection.

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

For part c, use the $\displaystyle \chi$ function: $\displaystyle \chi _E (x) = \left\{ {\begin{array}{lr} 1 & {x \in E} \\ 0 & {x \notin {\rm E}} \\ \end{array}} \right.$
• Jan 24th 2008, 03:17 PM
Jhevon
Quote:

Originally Posted by kuntah
I need to prove this plz help

$\displaystyle A,B,C \ and\ D\ are\ non\ empty\ sets\ with\ \left|A\right|=\left|B\right| and\ \left|C\right|=\left|D\right|$
$\displaystyle 1. Prove\ that\ \left|A\times C\right|=\left|B \times D\right|$

Hint: $\displaystyle |A \times C| = |A||C|$

Quote:

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

Quote:

3.$\displaystyle \left|P\left(A\right)\right|=\left|P\left(B\right) \right|$
With P i mean the power set
Hint: $\displaystyle |P(A)| = 2^{|A|}$

EDIT: ...ok, so both JaneBennet and Plato have different methods from mine, and theirs are similar, so chances are what I did was foolishness :D. you may ignore this post. i'll leave it up for anyone to comment on it.
• Jan 24th 2008, 04:00 PM
ThePerfectHacker
Quote:

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 :D. 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.
• Jan 24th 2008, 04:04 PM
Jhevon
Quote:

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