Question

Let $X=\{1,2,3,4,5\}$ and $Y=\{1,3,5,7,9\}$ . Which of the following is/are relations from $X$ to $Y$?

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

Answer 1

Answer: (B), (C)

$R_2=\left\{\right(1,1),(2,1),(3,3),(4,3),(5,5\left)\right\}$
$R_3=\left\{\right(1,1),(1,3),(3,5),(3,7),(5,7\left)\right\}$

$R_1=\left\{\right(x,y):y=2+x,x\in X,y\in Y\}$

$x=1\to y=1+2=3$

$x=2\to y=2+2=4$

$x=3\to y=3+2=5$

$x=4\to y=4+2=6$

$x=5\to y=5+2=7$

$R=\left\{\right(1,3),(2,4),(3,5),(4,6),(5,7\left)\right\}$

Here in$(4,6)$ ,$6$ does not belong to either $XorY$

So $R_1$ is not a relation between $X$ and $Y$

In $R_4$, since it contains an element $(7,9)$ which relates set Y to set Y, while rest elements relate set X to Y.

$\therefore R_4$ is not a relation between $X$ and $Y$