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I'm really confused on how to approach this question:

Recall that the Cartesian product $A \times A$ is defined as the set $\{(x, y) : x \in A \land y \in A \}$. Thus if for example $A = \{1, 2, 3\}$, $$A \times A = \{(1, 1),(1, 2),(1, 3),(2, 1),(2, 2),(2, 3),(3, 1),(3, 2),(3, 3)\}\;.$$ Consider a set $A \ne \varnothing$ where the number $|A|$ of elements of $A$ is $20$ less than the number $|A\times A|$ of elements in $A\times A$. Thus $|A| + 20 = |A\times A|$. Determine the number of elements in $A$.

Thank you in advance!

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Do you know how to find $|A\times B|$ if you know $|A|$ and $|B|$? –  Brian M. Scott Apr 14 '12 at 8:06
A further hint. Look at their examples of the sets $A$ and $A\times A$. –  Mike Apr 14 '12 at 8:09
I think the number of elements in AxA is 9? but i really get confused at the bit where "the number elements of A is 20 less than the number |A×A|". –  Xabi Apr 14 '12 at 8:26
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3 Answers

up vote 1 down vote accepted

Look at an example. Suppose that $A=\{1,2,3,4,5\}$. Then the members of $A\times A$ are:

$$\begin{array}{r|cc} &1&2&3&4&5\\ \hline 1&(1,1)&(1,2)&(1,3)&(1,4)&(1,5)\\ 2&(2,1)&(2,2)&(2,3)&(2,4)&(2,5)\\ 3&(3,1)&(3,2)&(3,3)&(3,4)&(3,5)\\ 4&(4,1)&(4,2)&(4,3)&(4,4)&(4,5)\\ 5&(5,1)&(5,2)&(5,3)&(5,4)&(5,5) \end{array}$$

Clearly $|A\times A|=5\cdot 5=25$. What happens in general? What is $|A\times A|$ in terms of $|A|$?

For future use, you should probably think about how $|A\times B|$ is determined by $|A|$ and $|B|$. You can use the same geometric idea. For instance, if $B=\{1,2,3\}$, with $A$ as in the previous example, the members of $A\times B$ are:

$$\begin{array}{r|cc} &1&2&3\\ \hline 1&(1,1)&(1,2)&(1,3)\\ 2&(2,1)&(2,2)&(2,3)\\ 3&(3,1)&(3,2)&(3,3)\\ 4&(4,1)&(4,2)&(4,3)\\ 5&(5,1)&(5,2)&(5,3) \end{array}$$

Clearly there are $5\cdot 3=15$ of them. What happens in general?

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Thank you for explaining it in a clear and concise way! ive managed to get { {1}, {2}, {3} }, { {1, 2}, {3} }, { {1, 2, 3} }, { {1, 3}, {2} }, { {1}, {2, 3} }, Which are all the elements in A. 5x5 = 25 5+20=25. –  Xabi Apr 14 '12 at 8:45
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How many elements are in $|A\times A|$ in terms of $n=|A|$? Plug this in and you'll have an equation to solve in terms of $n$.

Hint: Imagine a table where both rows and columns are indexed by elements of the set $A$. In each component of the table - namely the row indexed by $a\in A$ and $b\in A$ we can put $(a,b)\in A\times A$, thus giving a full listing of all of the elements of the Cartesian product.

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The number of elements in A is 5.

|A X A| = |A|.|A|

|A| + 20 = |A X A|
|A| + 20 = |A|.|A|

and solve this equation. you will get answer.

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Thank you ! 25=5x5 5+20=25 5+20 =5x5 ? –  Xabi Apr 14 '12 at 8:50
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