Question

Let the set A = {p, q, r}Define in extension A²Calculate Card A²Check that Card A² = (Card A) ²Card : Cardinality

Answer 1

Giving The set

`A=\mleft\lbrace p,q,r\mright\rbrace`

Then from cartesian coordinate of a set,

`A^2=A* A=\mleft\lbrace p,q,r\mright\rbrace*\mleft\lbrace p,q,r\mright\rbrace`

Then we expand the set above

`A^2=\mleft\lbrace(p,p\mright),(p,q),(p,r),(q,p),(q,q),(q,r),(r,p),(r,q),(r,r)\}`

The extension of a set is the set itself.

Hence

`\begin{gathered} \text{extension A}^2=A^2 \\ \text{extension A}^2=\lbrace(p,p),(p,q),(p,r),(q,p),(q,q),(q,r),(r,p),(r,q),(r,r)\} \end{gathered}`

Extension A² = {(p,p),(p,q),(p,r),(q,p),(q,q),(q,r),(r,p),(r,q),(r,r)}

The cardinality of a set is the number of elements in the set.

`\begin{gathered} \text{Cardinality of A}^2=cardA^2 \\ \text{cardA}^2=9 \end{gathered}`

Hence

Card A²=9

Then Card A is

`\begin{gathered} \sin ce\text{ A=}\mleft\lbrace p,q,r\mright\rbrace \\ \text{cardinality of A=CardA=3} \\ \text{Hence } \\ (CardA)=3^2=9^{} \end{gathered}`

Therefore

`\text{cardA}^2=(\text{card)}^2=9`

Therefore

cardA²=(card)²=9