Question

If $X=\{1,2,3,4,5\}$ and $Y=\{1,3,5,7,9\}$ then which of the following sets are relation from $X$ to $Y$

• A. $R_1=\left\{\right(x,a):a=x+2,x\in X,a\in Y\}$
• B. $R_2=\left\{\right(1,1),(2,1),(3,3),(4,3),(5,5\left)\right\}$
• C. $R_3=\left\{\right(1,1),(1,3),(3,5),(3,7),(5,7\left)\right\}$
• D. $R_4=\left\{\right(1,3),(2,5),(4,7),(5,9),(3,1\left)\right\}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $R_2=\left\{\right(1,1),(2,1),(3,3),(4,3),(5,5\left)\right\}$
C $R_3=\left\{\right(1,1),(1,3),(3,5),(3,7),(5,7\left)\right\}$
D $R_4=\left\{\right(1,3),(2,5),(4,7),(5,9),(3,1\left)\right\}$

$R_1=\{(1,3),(2,4),(3,5),(4,6),(5,7)\}$
since $4$ and $6$ do not belong to $y$.
$\therefore (2,4),(4,6)\notin R_1$
$\therefore R_1=\{(1,3),(3,5),(5,7)\}\subset X\times Y$
Hence, $R_1$ is a relation but not a mapping as the elements $2$ and $4$ do not have any image.
$R_2:$ It is certainly a mapping and since every mapping is a relation, it is a relation as well.
$R_3:$ It is a relation being a subset of $X\times Y$.
$R_4:$ It is both mapping and a relation. Each element in $X$ has a unique image. It is also one-one and on-to mapping and hence a bijection.