1.

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.

Answer»

Given is:

R = {(1, 2), (2, 3)} on the set A.

A = {1, 2, 3}

Right now, we have

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

Symmetric Relation:

We know (1, 2) ∈ R

Then, (2, 1) ∈ R

Also, (2, 3) ∈ R

Then, (3, 2) ∈ R

So, add (2, 1) and (3, 2) in R, so that we get

R’ = {(1, 2), (2, 1), (2, 3), (3, 2)}

Transitive Relation:

We need to make the relation R’ transitive.

So, we know (1, 2) ∈ R and (2, 1) ∈ R

Then, (1, 1) ∈ R

Also, (2, 3) ∈ R and (3, 2)

Then, (2, 2) ∈ R

Also, (2, 1) ∈ R and (1, 2) ∈ R

Then, (2, 2) ∈ R

Also, (3, 2) ∈ R and (2, 3) ∈ R

Then, (3, 3) ∈ R

Add (1, 1), (2, 2) and (3, 3) in R’, we get

R’’ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

Thus, we have got a relation which is reflexive, symmetric and transitive.

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

The ordered pair added are (1, 1), (2, 2), (3, 3), (3, 2).


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