Question

If R is a relation on the set A={ 1, 2, 3 } given by R={ (1, 1), (2, 2), (1, 3) } then R is NO SPAMMING PLS

Answer 1

Answer:

reflexive, symmetric, transitive.

Explanation:

The given relation is reflexive as for all 1,2,3∈A, (1,1),(2,2),(3,3)∈R.

Again the relation is symmetric as for all (1,1),(2,2),(3,3)∈R gives (1,1),(2,2),(3,3)∈R.

Again the relation is transitive also

Answer 2

Given set, A = {1, 2, 3}

Given relation, R = {(1, 1), (2, 2), (3, 3)}

Reflexive relation

If all the ordered pair elements of set A are in R then R is known to be called a reflexive relation.

Therefore, (a, a) $ \in $R, for all a$ \in $A

Where, a = a

So in set (A) all ordered pairs are (1, 1), (2, 2) and (3, 3) so all these ordered pairs are in set R so R is a reflexive relation.

Symmetric relation

If a, b $ \in $A such that (a, b) $ \in $ R then (b, a) $ \in $ R so this is called a symmetric relation.

In this case if a = b then this condition is also satisfied.

$ \Rightarrow a = b$

$ \Rightarrow b = a$

$ \Rightarrow \left( {b,a} \right) \in R$ For all a, b$ \in $A

So, R is also a symmetric relation.

Transitive relation

If a, b, c $ \in $A such that (a, b) $ \in $ R and (b, c) $ \in $ R then (a, c) $ \in $ R so this is called a transitive relation.

In this case if a = b = c then this condition is also satisfied.

$ \Rightarrow a = b,b = c$

$ \Rightarrow a = c$

$ \Rightarrow \left( {a,c} \right) \in R$ For all a, b, c$ \in $A

So, R is also a transitive relation.

Now as we all know if any relation R satisfying reflexive, symmetric and transitive then it is called as an equivalence relation.

So relation R is an equivalence relation.