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# Check Whether the Relation R Defined in the Set {1, 2, 3, 4, 5, 6} As R = {(A, B): B = A + 1} is Reflexive, Symmetric Or Transitive. - CBSE (Commerce) Class 12 - Mathematics

#### Question

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(ab): b = a + 1} is reflexive, symmetric or transitive.

#### Solution 1

Let A = {1, 2, 3, 4, 5, 6}.

A relation R is defined on set A as:

R = {(ab): b = a + 1}

∴R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

We can find (aa) ∉ R, where ∈ A.

For instance,

(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R

∴R is not reflexive.

It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.

∴R is not symmetric.

Now, (1, 2), (2, 3) ∈ R

But,

(1, 3) ∉ R

∴R is not transitive

Hence, R is neither reflexive, nor symmetric, nor transitive.

#### Solution 2

Reflexivity:

Letabeanarbitraryelementof R.Then,

1 cannot be true for all ∈ A.

⇒ (a, a∉ R

So, R is not reflexive on A.

Symmetry :

Let (a, b∈ R

⇒ 1

⇒ 1

⇒ − 1

Thus, (b, a∉ R

So, R is not symmetric on A.

Transitivity :

Let (1, 2) and (2, 3∈ R

⇒ 1 and 3 1  is true.

But 3  1+1

⇒ (1, 3∉ R

So, R is not transitive on A.

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Solution Check Whether the Relation R Defined in the Set {1, 2, 3, 4, 5, 6} As R = {(A, B): B = A + 1} is Reflexive, Symmetric Or Transitive. Concept: Types of Relations.
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