Question

Let $X=\{1,2,3,4\}$ and $Y=\{1,2,3,4\}$. Which of the following is a relation 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

The correct option is A $R_1=\left\{\right(x,y)|y=2+x,x\in X,y\in Y\}$

Given $X=\{1,2,3,4\}$ and $Y=\{1,2,3,4\}$

$X\times Y=\left\{\right(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)$

$(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)\}$
Now, $R_1=\left\{\right(1,3),(2,2\left)\right\}$

Since, $R_1\subset X\times Y$

Hence, $R_1$ is a relation from $X$ to $Y$
$R_2=\left\{\right(1,1),(2,1),(3,3),(4,3),(5,5\left)\right\}$

Since, $(5,5)\in R_2$ but $\notin X\times Y$

So, $R_2$ is not a relation from $X$ to$Y$
$R_3=\left\{\right(1,1),(1,3),(3,5),(3,7),(5,7\left)\right\}$

Since, $(3,5),(3,7),(5,7)\in R_3$ but $\notin X\times Y$

So, $R_3$ is not a relation from $X$ to $Y$
$R_4=\left\{\right(1,3),(2,5),(2,4),(7,9\left)\right\}$

Since, $(2,5),(7,9)\in R_4$ but $\notin X\times Y$

So, $R_4$ is not a relation from $X$ to $Y$