Let ,$A=\{a, b\}$ and $A^2$ is the set of all words of length 2 i.e., $A^2=A x A$.

(a). Find the element of $\mathrm{A}^2$.

(b). The relation $\mathrm{R}$ on $\mathrm{A}^2$ is defined by $\mathrm{x} \mathrm{Ry}=$ first letter in $\mathrm{x}$ is same as first letter in $\mathrm{y}$ when $\mathrm{x}$ and $\mathrm{y} \in \mathrm{A}^2$. Write $\mathrm{R}$ as a set of ordered pairs. Write domain and range.

Solution:

$ \begin{aligned}\\ A=&\{a, b\} \\\\ A^2=& A \times A \\\\ \end{aligned}\\ $

$ \begin{aligned}\\ (a) \quad A^2=? \\\\ A^2=&\{a, a\},\{a, b\},\{b, b\},\{b, a\} \\\\ \end{aligned}\\ $

(b) $R=(a b, a a, b a, b b)$

$ R=(a a, a b),(a b, a a),(b a, b b),(b b, b a) $

$ \begin{aligned}\\ &\text { Domain }=\{a a, a b, b a, b b\} \\\\ &\text { Range }=\{a b, a a, b b, b a\}\\ \end{aligned}\\ $