Question

If R is a relation in a set A=\left\\{ 1,2,3,4,5 \right\\} given by R=\left\\{ \left( a,b \right):\left| a-b \right|is\,even \right\\}, then consider the following statements.
I. R is an equivalence relation.
II. All elements of \left\\{ 1,3,5 \right\\} are related to each other and all elements of \left\\{ 2,4 \right\\} are related to each other.
III. All elements of \left\\{ 1,3,5 \right\\} are related to all elements of \left\\{ 2,4 \right\\}.
Choose the correct option.
(a) Only I is true.
(b) Only III is true.
(c) Both I and II are true.
(d) Neither I nor II is true.

Answer 1

Hint: Think of the basic definition of the types of relations given in the figure and try to check whether the relation mentioned in the questions satisfies the condition for any type of relation or not.

Complete step-by-step solution -
In the given question, we are given a set A=\left\\{ 1,2,3,4,5 \right\\}. On this set a relation is defined as, R=\left\\{ \left( a,b \right):\left| a-b \right|is\,even \right\\}.
Now, we know that the difference of two numbers even only when both the numbers are either even or both are odd.
Therefore, $\left| a-b \right|$ will be even, only when both a and b are either even or both a and b are odd.
Here, a, b belongs to set A. Therefore, possible even ordered pairs are: $\left( 2,2 \right),\left( 2,4 \right),\left( 4,2 \right),\left( 4,4 \right)$. And, possible odd ordered pairs are: $\left( 1,1 \right),\left( 1,3 \right),\left( 1,5 \right),\left( 3,1 \right),\left( 3,3 \right),\left( 3,5 \right),\left( 5,1 \right),\left( 5,3 \right),\left( 5,5 \right)$
Here, R=\left\\{ \left( a,b \right):\left| a-b \right|is\,even \right\\}
\left\\{ \left( 1,1 \right),\left( 1,3 \right),\left( 1,5 \right),\left( 2,2 \right),\left( 2,4 \right),\left( 3,1 \right),\left( 3,3 \right),\left( 3,5 \right),\left( 4,2 \right),\left( 4,4 \right),\left( 5,1 \right),\left( 5,3 \right),\left( 5,5 \right) \right\\}
Here, for all elements 1,2,3,4 and 5 of A, $\left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right),\left( 4,4 \right),\left( 5,5 \right)$ belongs to R. Therefore, R is reflexive.
Also, for all $\left( x,y \right)$ belongs to R, $\left( y,x \right)$ also belongs to R as can be clearly seen in tabular form of R. Therefore, R is symmetric.
Also, as we can clearly see in tabular form of R, for all $\left( x,y \right)$ and $\left( y,z \right)$ belongs to R, $\left( x,z \right)$ also belongs to R. therefore, R is transitive.
Hence, R is reflexive, symmetric and transitive. That is, R is an equivalent relation.
Therefore, Statement I is true.
Also, we can see clearly in R that all elements of \left\\{ 1,3,5 \right\\} are related to itself., that is $\left( x,y \right)$ such that x and y belongs to \left\\{ 1,3,5 \right\\}, belongs to R. The set which is formed is-
S = \left\\{ \left( 1,1 \right),\left( 1,3 \right),\left( 1,5 \right),\left( 3,1 \right),\left( 3,3 \right),\left( 3,5 \right),\left( 5,1 \right),\left( 5,3 \right),\left( 5,5 \right) \right\\}, we can see that R contains all the elements in the set S, hence S is a subset of R and all the elements in {1,3,5} are related to themselves.
Similarly, all elements of \left\\{ 2,4 \right\\} are related to itself, that is ordered pair $\left( x,y \right)$ belong to R where x and y belongs to \left\\{ 2,4 \right\\}. The set can be written as-
T = \left\\{ \left( 2,2 \right),\left( 2,4 \right),\left( 4,2 \right),\left( 4,4 \right) \right\\}
Therefore, II is true.
Now, \left\\{ 1,3,5 \right\\} is related to \left\\{ 2,4 \right\\} means that ordered pair $\left( x,y \right)$ belongs to R where x belongs to \left\\{ 1,3,5 \right\\} and y belongs to \left\\{ 2,4 \right\\}. But as can be clearly seen in tabular form of R, that is not true. The set formed is-
U = \left\\{ \left( 1,2 \right),\left( 1,4 \right),\left( 3,2 \right),\left( 3,4 \right),\left( 5,2 \right) ,\left( 5,4 \right) \right\\}
We can see that the elements of U are not present in R, so the elements are not related. Hence III is not true.
Therefore, the correct answer is option (c).

Note: If $\left( x,x \right)$ is present for some x in a set but not for all elements of the set, then the relation is not reflexive. So, don’t get confused. Even if there is one exception, the relation is not reflexive, symmetric, transitive or so on.