Let R and S be two equivalence relations on a set A.
Then,
a)
R ∪ S is an equivalence relation on A
b)
R ∩ S in an equivalence relation on A
c)
R - S is an equivalence relation on A
d)
None of the above
Correct answer is option 'B'. Can you explain this answer?

Question

Let R and S be two equivalence relations on a set A.
Then,

a)
R ∪ S is an equivalence relation on A
b)
R ∩ S in an equivalence relation on A
c)
R - S is an equivalence relation on A
d)
None of the above

Correct answer is option 'B'. Can you explain this answer?

Answer

Let R and S be two equivalence relations on a set A.

a) R ∩ S is an equivalence relation on A:
To show that R ∩ S is an equivalence relation, we need to prove three properties: reflexivity, symmetry, and transitivity.

1) Reflexivity: For any element a ∈ A, (a, a) must be in R ∩ S. Since R and S are both equivalence relations, (a, a) ∈ R and (a, a) ∈ S. Therefore, (a, a) ∈ R ∩ S, and R ∩ S is reflexive.

2) Symmetry: For any elements a, b ∈ A, if (a, b) ∈ R ∩ S, then (a, b) ∈ R and (a, b) ∈ S. Since R and S are both equivalence relations, (b, a) ∈ R and (b, a) ∈ S. Therefore, (b, a) ∈ R ∩ S, and R ∩ S is symmetric.

3) Transitivity: For any elements a, b, c ∈ A, if (a, b) ∈ R ∩ S and (b, c) ∈ R ∩ S, then (a, b) ∈ R and (a, b) ∈ S, and (b, c) ∈ R and (b, c) ∈ S. Since R and S are both equivalence relations, (a, c) ∈ R and (a, c) ∈ S. Therefore, (a, c) ∈ R ∩ S, and R ∩ S is transitive.

Since R ∩ S is reflexive, symmetric, and transitive, it is an equivalence relation on A.

b) R ∪ S is not necessarily an equivalence relation on A:
To show that R ∪ S is not necessarily an equivalence relation, we need to find a counterexample.

Let's consider a set A = {1, 2, 3}, and two equivalence relations R = {(1, 1), (2, 2), (3, 3)} and S = {(2, 2), (3, 3)}. The union of R and S is R ∪ S = {(1, 1), (2, 2), (3, 3)}.

Although R ∪ S is reflexive and symmetric, it is not transitive. For example, (1, 1) ∈ R ∪ S and (1, 1) ∈ R ∪ S, but (1, 1) ∉ R ∪ S. Therefore, R ∪ S is not an equivalence relation on A.

In conclusion, R ∩ S is an equivalence relation on A, while R ∪ S is not necessarily an equivalence relation on A.

Similar Mathematics Doubts

Let M be the set of men and R is a relation ‘is son of’ defined on M. Then, R is

Let R be an equivalence relation on a finite set A having n elements. Then, the number of ordered pairs in R is

Letthe set of natural numbers, define a relation on is eventhen

Let N be the set of integers. A relation R on N is defined as R = {(x, y) | xy > 0, x, y ∈ N}. Then, which one of the following is correct?

Let R be a relation over the set N x N and it is defined by (a, b)R (c, d) ⇒ a + d = b + c. Then R is

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams