Question

Let ‘R’ be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a,): both ‘a’ and ‘b’ are either odd or even}. Show that ‘R’ is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other but no element of the subset {1, 3, 5, 6} is related to any element of the subset {2, 4, 6}.

Answer 1

Hint:-For solving these questions, we would be requiring knowledge about symmetric, reflexive and transitive functions.

Complete step-by-step answer:
A relation is a relationship between sets of values. In math, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.
The types of relations are as follows
Reflexive Relation
A relation is a reflexive relation if every element of set A maps to itself. i.e. for every a \[\in \]
 A, (a, a) \[\in \] R.
Symmetric Relation
A symmetric relation is a relation R on a set A if (a, b) \[\in \] R then (b, a) \[\in \] R, for all a and b \[\in \] A.
Transitive Relation
If (a, b) \[\in \] R, (b, c) \[\in \] R, then (a, c) \[\in \] R, for all a, b, c \[\in \] A and this relation in set A is transitive.
Equivalence Relation
If and only if a relation is reflexive, symmetric and transitive, it is called an equivalence relation.
As mentioned in the question, we have to show that the given relation is an equivalence relation.
Now, we can define the function as follows
 R= {(a, b):a and b are either odd or even}
For checking the reflexivity, we can write the following
Now if we take any number \[a\in A\] , we can write as follows
So, a is either odd or even
(Because set A contains only odd and even numbers)
⇒ (a, a) \[\in \] R.
Hence, we can say that R is reflexive.
 For checking the symmetry, we can write the following
Now, if we take two different numbers a and b from set A, then we can write the following
a and b are either even or odd.
(Because they are elements of set A)
⇒ (a, b) \[\in \] R and (b, a) \[\in \] R.
Hence, we can say that R is symmetric.

For checking the transitivity, we can write the following
Now, if we take three different numbers a and b and c from set A, so, we can write the following
a, b and c are either even or odd.
⇒​ (a, b) \[\in \] R, (b, c) \[\in \] R and (c, a) \[\in \] R.
Therefore, we can say that R is transitive.
​Hence, the given relation is an equivalence relation.
 Now, we can say that the elements of set {1, 3, 5, 7} can be related to each other due to the following example
R contains (1, 1), so it is satisfying its reflexivity.
R contains (1, 3) and R contains (3, 1), so it is satisfying its symmetry.
R contains (1, 3), R contains (3, 5) and R contains (5, 1), so it is satisfying its transitivity.
Hence, we can say that all the elements of set {1, 3, 5, 7} are related to each other.
Similarly,
We can say that the elements of set {2, 4, 6} can be related to each other due to the following example
R contains (2, 2), so it is satisfying its reflexivity.
R contains (2, 4) and R contains (4, 2), so it is satisfying its symmetry.
R contains (2, 4), R contains (4, 6) and R contains (6, 2), so it is satisfying its transitivity.
Hence, we can say that all the elements of set { 2, 4, 6} are related to each other.
Now, for relation between set {1, 3, 5, 7} and set { 2, 4, 6}, we can write as follows
R does not contain any element that satisfies reflexivity of the relation from set {1, 3, 5, 7} to set { 2, 4, 6}.
Hence, no elements from set {1, 3, 5, 7} are related to set { 2, 4, 6}.

Note: -The students can make an error if they don’t know about the definitions and the meaning of different types of relations.
Also, it is important to know about the different types of set representations that are the rooster form or the set builder form as without knowing these one could never understand the question properly.