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

Answer 1

Answer: (B), (C)

$R_3=\{(1,1),(1,3),(3,5),(3,7),(5,7)\}$

Explanation for the correct answer:

$R1=\left\{\right(x,y\left)\right|y=2+x,x\in Y,y\in Y\}$

If $x$ takes different, the values of $y$ also differ accordingly.

$$\begin{gathered} x=1\to y=1+2=3 \\ x=2\to y=2+2=4 \\ x=3\to y=3+2=5 \\ x=4\to y=4+2=6 \\ x=5\to y=5+2=7 \\ R=(1,3),(2,4),(3,5),(4,6),(5,7) \end{gathered}$$

In $(4,6),6\notin$ either $X$ or $Y$

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

In $R_4$, the element $(7,9)$ is present which relates set $Y$ to set $Y$, while the other elements relate set $X$ to$Y$.

$R_4$ is not a relation between $X$ and$Y$.

Hence, the correct option are (B) and (C)