# Can a relation be transitive if it is not symmetric?

## Can a relation be transitive if it is not symmetric?

xTy iff x > y + 1 , is clearly non symmetric and not reflexive. Suppose y > z + 1; we deduced x > y + 1 > z + 2 > z + 1, hence T is transitive.

### How can a relation be symmetric and transitive but not reflexive?

(b) The relation R2 = {(1,2),(2,1),(2,2),(1,1)} is symmetric and transitive, but not reflexive. Relation R2 is not reflexive because 3 R 2 3. Relation R2 is symmetric because the only a, b ∈ A with a = b for which aR2 b is a = 1 or 2 and b = 1 or 2. Since 1R2 2 and 2R2 1, R2 is symmetric.

#### Is a relation reflexive if it is symmetric and transitive?

Suppose R is a relation on A. If R is symmetric and transitive, then R is reflexive. False proof.

**Can a relation be symmetric and not reflexive?**

Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R. Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric. ∴R is not transitive. Hence, relation R is symmetric but not reflexive or transitive.

**Which is an example of a relation which is reflexive transitive but not symmetric?**

Solution : “The relation `x ge y` on z” is reflexxive , transitive but not symmetric.

## Is reflexive also symmetric?

If a relation is reflexive, symmetric and transitive it is an equivalence relation. This means that it splits the base set into disjoint subsets (equivalence classes) in which every element is related to itself and every other element in the class to which it belongs.

### Which of the following relations is symmetric but neither reflexive nor transitive for a set a p q R a R {( p q p R p/s )} B R {( p q/q p )} c/r {(?

Hence, relation R is symmetric and transitive but not reflexive.

#### Are all reflexive relation symmetric?

No, it doesn’t. A relation can be symmetric and transitive yet fail to be reflexive. Say you have a symmetric and transitive relation on a set , and you pick an element .

**Is every reflexive relation is symmetric?**

No, you’re only considering the diagonal of the set, which is always an equivalence relation.

**Are all reflexive relations transitive?**

Yes. Such a relation is indeed a transitive relation, since the only relevant cases for the premise “xRy∧yRz” are x=y=z in such relations. Since the premise never holds for cases where x,y,z are not all the same, there is no need to consider them.

## Which of the following relations is symmetric but neither reflexive nor transitive on a set A ={ 1 2 3?

### Which of the following relation is symmetric and transitive but not reflexive for the set i 4 5?

Explanation: R= {(4, 5), (5, 4), (4, 4)} is symmetric since (4, 5) and (5, 4) are converse of each other thus satisfying the condition for a symmetric relation and it is transitive as (4, 5)∈R and (5, 4)∈R implies that (4, 4) ∈R. It is not reflexive as every element in the set I is not related to itself.