# Power sets and Cartesian products.

• cgjolberg
In summary, the statement "For every pair of sets (A,B), P(AxB)=P(A)xP(B) is not always true. A counterexample can be shown by choosing sets A={1,2} and B={a,b} and comparing the sizes of P(AxB) and P(A)xP(B). Therefore, P(AxB) and P(A)xP(B) are not equivalent statements.
cgjolberg

## Homework Statement

For every pair of sets (A,B) we have P(AxB)=P(A)xP(B)
Prove or disprove the above statement.

## The Attempt at a Solution

I have attempted solving this using A={1,2} and B={a,b}
AxB={(1,a),(1,b),(2,a),(2,b)}
P(AxB)={ø,{(1,a)},{(1,b)},{(2,a)},{(2,b)},{(1,a),(1,b)},{(1,a),(2,a)},{(1,a),(2,b)},{(1,b),(2,a)},{(1,b),(2,b)},{(2,a),(2,b)},{(1,a),(1,b),(2,a)},{(1,a),(1,b),(2,b)},{(1,a),(2,a),(2,b)},{(1,b),(2,a),(2,b)},{(1,a),(1,b),(2,a),(2,b)}}

P(A)={ø,{1},{2},{1,2}}
P(B)={ø,{a},{b},{a,b}}
P(A)xP(B)={(ø,ø),(ø,{a}),(ø,{b}),(ø,{a,b}),({1},ø),({1},{a}),({1},{b}),({1},{a,b}),({2},ø),({2},{a}),({2},{b}),({2},{a,b}),({1,2},ø),({1,2},{a}),({1,2},{b}),({1,2},{a,b})}

So from this it seems clear that they are not equal. The easiest way to state that without all of the work I just did seems to be that P(AxB) creates sets of sets, whearas P(A)xP(B) creates pairs of sets or something along those lines.
However, from several websites I have found through searching, several say these two statements are equivalent, so now I am confused.

Also, am I correct in crossing the ø through like I did in P(A)xP(B)? I am not sure if you are suppose to do it like I did or just have one ø at the beginning of the answer.
Thanks for the help!

What you did was correct.

You can also use a combinatorial argument to disprove.

Let |A| denote the order of A.

Suppose ##|A| = m## and ##|B| = n##. Then ##|P(A)| = 2^m## and ##|P(B)| = 2^n##.

##|A \times B| = mn ## so ##|P(A \times B)| = 2^{mn}##.

But ##|P(A) \times P(B)| = 2^m*2^n = 2^{m+n}##.

So if ##mn \neq m + n## they have different sizes.

Funny enough, you pick m and n to be 2 so they have the same size. But your example is perhaps more insightful because you see that the objects in these sets are fundamentally different.

## 1. What is a power set?

A power set is a set that contains all the possible subsets of a given set. It includes the empty set and the original set itself.

## 2. How is a power set denoted?

A power set is usually denoted by 2n, where n is the number of elements in the original set.

## 3. What is the cardinality of a power set?

The cardinality of a power set is 2n, where n is the number of elements in the original set. This means that a power set will always have more elements than the original set.

## 4. What is a Cartesian product?

A Cartesian product is a mathematical operation that combines two sets to create a new set. The resulting set consists of all possible ordered pairs where the first element comes from the first set and the second element comes from the second set.

## 5. How is a Cartesian product denoted?

A Cartesian product is denoted by A x B, where A and B are the two sets being combined. The resulting set will contain all possible ordered pairs of elements from A and B.

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