Which Relation is Not a Function?
Question:
Let
$$
A=\{p,q,r,s\}
$$
and
$$
B=\{1,2,3\}
$$
Which of the following relations from \(A\) to \(B\) is not a function?
(i) $$ R_1=\{(p,1),(q,2),(r,1),(s,2)\} $$ (ii) $$ R_2=\{(p,1),(q,1),(r,1),(s,1)\} $$ (iii) $$ R_3=\{(p,1),(q,2),(p,2),(s,3)\} $$ (iv) $$ R_4=\{(p,2),(q,3),(r,2),(s,2)\} $$
(i) $$ R_1=\{(p,1),(q,2),(r,1),(s,2)\} $$ (ii) $$ R_2=\{(p,1),(q,1),(r,1),(s,1)\} $$ (iii) $$ R_3=\{(p,1),(q,2),(p,2),(s,3)\} $$ (iv) $$ R_4=\{(p,2),(q,3),(r,2),(s,2)\} $$
Solution
A relation is a function if every element of set \(A\) has exactly one image in set \(B\).
Checking \(R_1\)
Each element \(p,q,r,s\) has exactly one image.
Therefore, \(R_1\) is a function.
Checking \(R_2\)
Each element \(p,q,r,s\) has exactly one image.
Therefore, \(R_2\) is a function.
Checking \(R_3\)
Here, element \(p\) has two images:
$$ 1 \text{ and } 2 $$
Also, element \(r\) has no image.
Therefore, \(R_3\) is not a function.
Checking \(R_4\)
Each element \(p,q,r,s\) has exactly one image.
Therefore, \(R_4\) is a function.
Hence,
$$ \boxed{R_3 \text{ is not a function}} $$