Mathematics Question on Functions

Question

If R and R1 are equivalence relations on a set A, then so are the relations

Answer 1

Solution

The correct answer is/are option(s):
(A): R-1
(C): R∩R1

Given, R1 and R
are equivalence relations on set A.
A relation is called the equivalence relation if it is reflexive, symmetric and transitive.
Since, R1
is a transitive relation.
(1) R1
is reflexive.
i.e. (a,a)∈R1 for all a∈A
(2) R1
is symmetric.
i.e. if (a,b)∈R1, then (b,a)∈R1; a,b∈A
(3) R1
is transitive.
i.e. if (a,b)∈R1, and (b,c)∈R1 then (a,c)∈R1; a,b,c∈A
Similarly, R2
is also an equivalence relation. So,
(1) R
is symmetric.
i.e. if (a,b)∈R then (b,a)∈R; a,b∈A
(2) R
is reflexive.
i.e. (a,a)∈R for all a∈A
(3) R
is transitive.
i.e. if (a,b)∈R, and (b,c)∈R then (a,c)∈R; a,b,c∈A
We have to prove that R1∩R
is an equivalence relation.
Check reflexive:
For all a∈A,(a,a)∈R1 and (a,a)∈R [As both R1 and R2 are reflexive on A]
We know that if an element belongs to set A and also to set B, then the element will also belong to (A∩B).So, as a∈A,(a,a)∈R1 and (a,a)∈R(a,a)∈(R1∩R)

∴(R1∩R) is reflexive on A.