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. $\mathrm{R}_{1}=\{(x, y) \mid y=2+x, x \in Y, y \in Y\}$
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

Hint: In the given question, we are given sets. It belongs to the Cartesian product's subgroup. For this, we analyze each option one by one to check whether the given sets are relations from X to Y. for this, 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:
Given that
$X=\{1,2,3,4,5\}$ and $Y=\{1,3,5,7,9\}$
$R_{1}=\{(x, y): y=2+x, x \in X, y \in Y\}$
Relations and sets are related to one another. The connection explains how two sets are related.
If the element x is from the set X and y is from the second set Y, then the elements are said to be related if the ordered (x,y) in the relation.
$x=1 \rightarrow y=1+2=3$
$x=2 \rightarrow y=2+2=4$
$x=3 \rightarrow y=3+2=5$
$x=4 \rightarrow y=4+2=6$
$x=5 \rightarrow y=5+2=7$
$R = {(1, 3), (2, 4), (3, 5), (4, 6), (5, 7)}$
Here in $(4,6), 6$ does not belong to either X or $Y$
$\mathrm{R}_{1}$ is not a relation between $\mathrm{X}$ and $\mathrm{Y}$
For $R_{2}=\{(1,1),(2,1),(3,3),(4,3),(5,5)\}$
we see all the first terms in ordered pair 1,2,3,4,5 belong to X and all the second terms in ordered pair 1,1,3,3,5 belong to Y. Hence, all ordered pair belong to relation from X to Y.
For $R_{3}=\{(1,1),(1,3),(3,5),(3,7),(5,7)\}$
we see all the first terms in ordered pair 1,1,3,3,5 belong to X and all the second terms in ordered pair 1,3,5,7,7 belong to Y. Hence, all ordered pair belong to relation from X to Y.
The element $(7,9)$ that connects set $\mathrm{Y}$ to set $\mathrm{Y}$ is present in $\mathrm{R}_{4}$, while the other elements connect set $\mathrm{X}$to set $Y$
$\mathrm{R}_{4}$ is not a relation between $\mathrm{X}$ and $\mathrm{Y}$

So the correct answer is option (B) and (C).

Note: A function is a relation that states that there should only be one output for each input. Alternatively, we could say that a function is a special sort of relation (a collection of ordered pairs) that adheres to the rule that every X-value should only be connected with one y-value.
Students should note that for relation from X to Y, every element of given relation should be of the form (x,y) where \[x\in X,y\in Y\]. It should not be confused with (y,x) where \[y\in Y,x\in X\].