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!
3 Answers
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.
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?
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.
