Question

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

• A. $R_1=\left\{\right(x,y)|y=2x+1,x\in X,y\in Y\}$
• 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: (A)

$R_1=\left\{\right(x,y)|y=2x+1,x\in X,y\in Y\}$

We have
$X=\{1,2,3,4\}$ and $Y=\{1,3,5,7,9\}$.
$X\times Y=\left\{\right(1,1),(1,3),(1,5),(1,7),(1,9),(2,1),(2,3),(2,5),$
$(2,7),(2,9),(3,1),(3,3),(3,5),(3,7),(3,9),(4,1),(4,3),(4,5),(4,7),(4,9)\}$
Let's take option A,
$R_1=\left\{\right(x,y)|y=2x+1,x\in X,y\in Y\}$
$R_1=\left\{\right(1,3),(2,5),(3,7),(4,9\left)\right\}$
Since, $R_1\subseteq X\times Y$
So, $R_1$ is a relation from $X$ to $Y$
Clearly, $R_2$ is not a relation from $X$ to $Y$ as $(5,5)\notin X\times Y$
Similarly, $R_3$ is not a relation from $X$ to $Y$ as $(5,7)\notin X\times Y$
In the same manner, $R_4$ is also not a relation from $X$ to $Y$ as $(7,9)\notin X\times Y$