Q.Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. A relation $R$ from $A$ to $B$ is defined by $R = \{(a, b) : b = a^{2},\ a \in A,\ b \in B\}$. Write $R$ in roster form and state its domain and range.

Apply the rule $b = a^{2}$ to each $a \in A$, keep the pairs whose second value actually lies in $B$, then read off the first and second components.

Apply the defining rule $b = a^{2}$ to each element $a$ of $A$, and include the pair only if the resulting $b$ belongs to $B$:

• $a = 1:\ b = 1^{2} = 1 \in B \Rightarrow (1, 1) \in R$
• $a = 2:\ b = 2^{2} = 4 \in B \Rightarrow (2, 4) \in R$
• $a = 3:\ b = 3^{2} = 9 \in B \Rightarrow (3, 9) \in R$
• $a = 4:\ b = 4^{2} = 16 \in B \Rightarrow (4, 16) \in R$

So in roster form

Domain = set of first components = $\{1, 2, 3, 4\}$. …

Students sometimes include a pair whose square is not in $B$ (if such a case arose), or confuse the domain (first components, from $A$) with the range (second components, …