Table of Contents

- 1 Does there exist a relation R on A that is both an equivalence relation and an order relation?
- 2 How do you prove that R is an equivalence relation?
- 3 What relation is both equivalence and partial order?
- 4 Can a relation be both an equivalence relation and a total order?
- 5 How many equivalence relations on the set 1/2 are there in all if the relation may or may not contain 1/2 justify your answer?
- 6 Is r1 union r2 equivalence relation?
- 7 When is $R\\cap s$ not an equivalence relation?
- 8 Is $R$ and $s$ reflexive symmetric and transitive?

## Does there exist a relation R on A that is both an equivalence relation and an order relation?

Yes the Identity relation is both Partial order and Equivalence. example: A={1,2,3,4}, R={(1,1),(2,2),(3,3),(4,4)} ,Here R is both Partial order and quivalence.

## How do you prove that R is an equivalence relation?

To prove R is an equivalence relation, we must prove R is reflexive, symmetric, and transitive. So let a, b, c ∈ R. Then a − a = 0=0 · 2π where 0 ∈ Z. Thus (a, a) ∈ R and R is reflexive.

**Is intersection of two equivalence relations itself an equivalence relation?**

The intersection of two equivalence relations is itself an equivalence relation.

**Is the union of two equivalence relations An equivalence relation?**

Hence the union of two equivalence relations is not necessarily an equivalence relation.

### What relation is both equivalence and partial order?

Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric.

### Can a relation be both an equivalence relation and a total order?

Clearly it is also transitive, and hence it is the only relation that is both a partial order and an equivalence relation. The same argument goes for any set S: The only relation that is both a partial order and an equivalence relation is the identity relation R={(x,x)∣x∈S}.

**Is the relation An equivalence relation?**

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation is equal to is the canonical example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes.

**Which of the following is true a union of two equivalence relations is an equivalence relation B intersection of two equivalence relations is an equivalence relation?**

3 Answers. No union of two equivalence relations may not be an equivalence relation because of transitive dependency. Union ={(1,2),(3,4),(1,4),(2,3)} which is not transitive i.e. (1,3) and (2,4) is missing.

## How many equivalence relations on the set 1/2 are there in all if the relation may or may not contain 1/2 justify your answer?

Transitive means if (a,b) is in relation and (b,c) is in relation, then (a,c) is in relation. So, if (1,2) is in relation and (2,1) is in relation, then (1,1) should be in relation. Hence, only two possible relations are there which are equivalence.

## Is r1 union r2 equivalence relation?

Thus R1 ∩ R2 is reflexive, symmetric and also transitive. Thus R1 ∩ R2 is an equivalence relation.

**Is R an equivalence relation or partial order?**

Here are a few special types of relations: Equivalence Relation: R is an equivalence relation iff it is reflexive, symmetric, and transitive. Partial Order: X is partially ordered by R (or R is a partial order on X) iff R is reflexive, anti-symmetric, and transitive.

**What is an equivalence relation give an example?**

Equivalence Relations. Definition. An equivalence relation on a set S, is a relation on S which is. reflexive, symmetric and transitive. Examples: Let S = ℤ and define R = {(x,y) | x and y have the same parity} i.e., x and y are either both even or both odd. The parity relation is an equivalence relation.

### When is $R\\cap s$ not an equivalence relation?

“Given $R$ and $S$ are equivalence relations on a set $A$, suppose for the sake of contradiction, that $R\\cap S$ is NOT an equivalence relation…”. If not an equivalence relation, then $R\\cap S$ fails to be reflexive and/or fails to be symmetric, and/or fails to be transitive.

### Is $R$ and $s$ reflexive symmetric and transitive?

Hint: Use the fact that $R$ and $S$ are EQUIVALENCE relations on THE SAME set, and hence both must be reflexive, symmetric, and transitive on that set.

**Can We say every empty relation is an equivalence relation?**

We can say that the empty relation on the empty set is considered as an equivalence relation. But, the empty relation on the non-empty set is not considered as an equivalence relation. Can we say every relation is a function? No, every relation is not considered as a function, but every function is considered as a relation.