Question

If $X=\{1,2,3,4,5\},Y=\{1,3,5,7,9\},$ then which among the following is not a relation from $X$ to $Y$

• A. $R_1=\left\{\right(x,y):y=x+2;x\in X,y\in Y\}$
• 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. $R_4=\left\{\right(1,3),(2,1\left)\right(2,5),(3,7),(7,9\left)\right\}$

Answer 1

The correct option is D $R_4=\left\{\right(1,3),(2,1\left)\right(2,5),(3,7),(7,9\left)\right\}$

We have,
$R_1=\left\{\right(x,y):y=x+2;x\in X,y\in Y\}$
$=\left\{\right(1,3),(3,5),(5,7\left)\right\}$
so, we can say that, $R_1,R_2,R_3$ are subsets of $X\times Y$.

In $R_4,$ we have $(7,9)$
but $(7,9)\notin X\times Y,$ because $7$ is not an element of set $X.$ So, $R_4$ is not a subset of $X$ to $Y$
$\therefore R_4$ is not a relation from $X$ to $Y$