Question

Let we have two sets $X=\left\{ 1,2,3,4,5 \right\}$ and $Y=\left\{ 1,3,5,7,9 \right\}$ which of the following is/are relations from X to Y?
\[\begin{align}
  & A.{{R}_{1}}=\left\{ \left( x,y \right):y=2+x,x\in X,y\in Y \right\} \\
 & B.{{R}_{2}}=\left\{ \left( 1,1 \right),\left( 2,1 \right),\left( 3,3 \right),\left( 4,3 \right),\left( 5,5 \right) \right\} \\
 & C.{{R}_{3}}=\left\{ \left( 1,1 \right),\left( 1,3 \right),\left( 3,5 \right),\left( 3,7 \right),\left( 5,7 \right) \right\} \\
 & D.{{R}_{4}}=\left\{ \left( 1,3 \right),\left( 2,5 \right),\left( 2,4 \right),\left( 7,9 \right) \right\} \\
\end{align}\]

Answer 1

Hint: In this question, we are given two sets and we have to determine relations from the given options. For this, we will analyze every option one by one to check whether given sets are relations from X to Y, the first element from every ordered pair should be from set X and the second element should be from set Y.

Complete step-by-step solution:
Let us first understand the meaning of the relation between two sets.
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set X and y is from the second set Y, then objects are said to be related if the ordered pair (x, y) is in the relation.
Now, let us analyze all the options one by one.
For set ${{R}_{1}}$ we are given set as,
\[{{R}_{1}}=\left\{ \left( x,y \right):y=2+x,x\in X,y\in Y \right\}\]
For x = 1, y = 2+1=3 ordered pair becomes (1,3).
For x = 2, y = 2+2=4 ordered pair becomes (2,4).
For x = 3, y = 3+2=5 ordered pair becomes (3,5).
For x = 4, y = 4+2=6 ordered pair becomes (4,6).
For x = 5, y = 5+2=7 ordered pair becomes (5,7).
Hence, \[{{R}_{1}}=\left\{ \left( 1,3 \right),\left( 2,4 \right),\left( 3,5 \right),\left( 4,6 \right),\left( 5,7 \right) \right\}\]
For pairs (2,4) and (4,6), 4 and 6 do not belong to Y. Hence, they do not belong to relation from X to Y. So, ${{R}_{1}}$ is not a relation from X to Y.
For ${{R}_{2}}$ we are given set as,
\[{{R}_{2}}=\left\{ \left( 1,1 \right),\left( 2,1 \right),\left( 3,3 \right),\left( 4,3 \right),\left( 5,5 \right) \right\}\]
As we can see all first terms in ordered pairs 1, 2, 3, 4, 5 belong to X and all second terms in ordered pairs 1, 1, 3, 3, 5 belong to Y. Hence, all ordered pairs belong to relation from X to Y. So, ${{R}_{2}}$ is a relation from X to Y.
For ${{R}_{3}}$ we are given set as,
\[{{R}_{3}}=\left\{ \left( 1,1 \right),\left( 1,3 \right),\left( 3,5 \right),\left( 3,7 \right),\left( 5,7 \right) \right\}\]
As we can see all the first terms in ordered pairs 1, 1, 3, 3, 5 belong to X and all second terms in ordered pairs 1, 3, 5, 7, 7 belong to Y. Hence, all ordered pairs belong to relation from X to Y.
So, ${{R}_{3}}$ is a relation from X to Y.
For ${{R}_{4}}$ we are given set as,
\[{{R}_{4}}=\left\{ \left( 1,3 \right),\left( 2,5 \right),\left( 2,4 \right),\left( 7,9 \right) \right\}\]
As we can see, in (7,9) 7 does not belong to X and 9 does not belong to Y. Therefore, it does not belong to relation from X to Y. Hence, ${{R}_{4}}$ is not a relation from X to Y.
Thus, options B and C are the relation from X to Y.

Note: Students should note that, for relation from X to Y, every element of given relation should be of the form (x, y) where $x\in X,y\in Y$. They should not confuse it with relation from Y to X where elements are of the form (y, x) where $y\in Y,x\in X$. For ${{R}_{1}}$ we have converted the set to roaster form so as to observe easily.