Relations

Question

If R and S are two equivalence relations on a set A, then which of the following is always an equivalence relation?

RankGuru · Accepted Answer

1. **Reflexive Check**: If $xRx$ and $xSx$, then $(x, x)$ must be in both $R$ and $S$. Thus $(x, x) \in R \cap S$. (Reflexive)
2. **Symmetric Check**: If $(x, y) \in R \cap S$, then $(x, y) \in R$ and $(x, y) \in S$.
   - By symmetry of $R, S$, both $(y, x) \in R$ and $(y, x) \in S$.
   - Thus $(y, x) \in R \cap S$. (Symmetric)
3. **Transitive Check**: If $(x, y) \in R \cap S$ and $(y, z) \in R \cap S$, then the pairs are in both $R$ and $S$.
   - By transitivity of $R, S$, $(x, z) \in R$ and $(x, z) \in S$.
   - Thus $(x, z) \in R \cap S$. (Transitive)
4. **Counter-example for Union**: If $R = \{(1,1), (2,2), (1,2), (2,1)\}$ and $S = \{(2,2), (3,3), (2,3), (3,2)\}$, then $R \cup S$ contains $(1,2)$ and $(2,3)$ but not $(1,3)$. Violates transitivity.

Relations

Select the correct option:

🔒 Solution Hidden from View

More sets, relations and functions Practice Questions

If R is a relation from a finite set A having m elements to a finite set B having n elements, then t...

Let R be a relation on the set N of natural numbers defined by nRm \u21Longleftrightarrow n is a fac...

The relation 'less than' on the set of real numbers R is

Let R be the relation on the set R of all real numbers defined by aRb \u21Longleftrightarrow |a - b|...

If A = {a, b, c}, then the number of equivalence relations on A containing (a, b) is

The minimum number of ordered pairs that must be added to the relation R = {(1, 2), (2, 3)} on the s...

Let R be a relation on the set N of natural numbers defined by R = {(x, y) : x + 2y = 8}. The range ...

A relation R on set A = {1, 2, 3, 4} is defined as R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)}. W...

The relation R = {(1,1), (2,2), (3,3)} on the set A = {1, 2, 3} is

Let R be a relation on Z (set of integers) defined by R = {(a, b) : a - b is divisible by 5}. Then R...

The number of reflexive relations on a set with 3 elements is

Let R be a relation from set A = {2, 3, 4, 5} to set B = {1, 4, 5} defined by R = {(x, y) : x divide...

About This Question

Solution