Q. 23.6( 12 Votes )

# Relations R_{1}, R_{2}, R_{3} and R_{4} are defined on a set A = {a, b, c} as follows :

R_{1} = {(a, a) (a, b) (a, c) (b, b) (b, c), (c, a) (c, b) (c, c)}

R_{2} = {(a, a)}

R_{3} = {(b, a)}

R_{4} = {(a, b) (b, c) (c, a)}

Find whether or not each of the relations R_{1}, R_{2}, R_{3,} R_{4} on A is (i) reflexive (iii) symmetric (iii) transitive.

Answer :

We have set,

A = {a, b, c}

Here, R_{1}, R_{2}, R_{3,} and R_{4} are the binary relations on set A.

So, recall that for any binary relation R on set A. We have,

R is reflexive if for all x ∈ A, xRx.

R is symmetric if for all x, y ∈ A, if xRy, then yRx.

R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

So, using these results let us start determining given relations.

We have

R_{1} = {(a, a) (a, b) (a, c) (b, b) (b, c) (c, a) (c, b) (c, c)}

(i). **Reflexive:**

For all a, b, c ∈ A. [∵ A = {a, b, c}]

Then, (a, a) ∈ R_{1}

(b, b) ∈ A

(c, c) ∈ A

[∵ R_{1} = {**(a, a)** (a, b) (a, c) **(b, b)** (b, c) (c, a) (c, b) **(c, c)**}]

So, ∀ a, b, c ∈ A, then (a, a), (b, b), (c, c) ∈ R.

**∴** **R _{1} is reflexive.**

(ii). **Symmetric:**

If (a, a), (b, b), (c, c), (a, c), (b, c) ∈ R_{1}

Then, clearly (a, a), (b, b), (c, c), (c, a), (c, b) ∈ R_{1}

∀ a, b, c ∈ A

[∵ R_{1} = {**(a, a) (a, b) (a, c) (b, b) (b, c) (c, a) (c, b) (c, c)**}]

But, we need to try to show a contradiction to be able to determine the symmetry.

So, we know (a, b) ∈ R_{1}

But, (b, a) ∉ R_{1}

So, if (a, b) ∈ R_{1}, then (b, a) ∉ R_{1}.

∀ a, b ∈ A

**∴** **R _{1} is not symmetric.**

(iii). **Transitive:**

If (b, c) ∈ R_{1} and (c, a) ∈ R_{1}

But, (b, a) ∉ R_{1} [Check the Relation R_{1} that does not contain (b, a)]

∀ a, b ∈ A

[∵ R_{1} = {(a, a) (a, b) (a, c) (b, b) **(b, c) (c, a)** (c, b) (c, c)}]

So, if (b, c) ∈ R_{1} and (c, a) ∈ R_{1}, then (b, a) ∉ R_{1}.

∀ a, b, c ∈ A

**∴** **R _{1} is not transitive.**

Now, we have

R_{2} = {(a, a)}

(i). **Reflexive:**

Here, only (a, a) ∈ R_{2}

for a ∈ A. [∵ A = {a, b, c}]

[∵ R_{2} = {**(a, a)**}]

So, for a ∈ A, then (a, a) ∈ R_{2}.

**∴** **R _{2} is reflexive.**

(ii). **Symmetric:**

For symmetry,

If (x, y) ∈ R, then (y, x) ∈ R

∀ x, y ∈ A.

Notice, in R_{2} we have

R_{2} = {**(a, a)**}

So, if (a, a) ∈ R_{2}, then (a, a) ∈ R_{2}.

Where a ∈ A.

**∴** **R _{2} is symmetric.**

(iii). **Transitive:**

Here,

(a, a) ∈ R_{2} and (a, a) ∈ R_{2}

Then, obviously (a, a) ∈ R_{2}

Where a ∈ A.

[∵ R_{2} = {**(a, a)**}]

So, if (a, a) ∈ R_{2} and (a, a) ∈ R_{2}, then (a, a) ∈ R_{2}, where a ∈ A.

**∴** **R _{2} is transitive.**

Now, we have

R_{3} = {(b, a)}

(i). **Reflexive:**

∀ a, b ∈ A [∵ A = {a, b, c}]

But, (a, a) ∉ R_{3}

Also, (b, b) ∉ R_{3}

[∵ R_{3} = {**(b, a)**}]

So, ∀ a, b ∈ A, then (a, a), (b, b) ∉ R_{3}

**∴** **R _{3} is not reflexive.**

(ii). **Symmetric:**

If (b, a) ∈ R_{3}

Then, (a, b) should belong to R_{3}.

∀ a, b ∈ A. [∵ A = {a, b, c}]

But, (a, b) ∉ R_{3}

[∵ R_{3} = {**(b, a)**}]

So, if (a, b) ∈ R_{3}, then (b, a) ∉ R_{3}

∀ a, b ∈ A

**∴** **R _{3} is not symmetric.**

(iii). **Transitive:**

We have (b, a) ∈ R_{3} but do not contain any other element in R_{3}.

Transitivity can’t be proved in R_{3}.

[∵ R_{3} = {**(b, a)**}]

So, if (b, a) ∈ R_{3} but since there is no other element.

**∴** **R _{3} is not transitive.**

Now, we have

R_{4} = {(a, b) (b, c) (c, a)}

(i). **Reflexive:**

∀ a, b, c ∈ A [∵ A = {a, b, c}]

But, (a, a) ∉ R_{4}

Also, (b, b) ∉ R_{4} and (c, c) ∉ R_{4}

[∵ R_{4} = {(a, b) (b, c) (c, a)}]

So, ∀ a, b, c ∈ A, then (a, a), (b, b), (c, c) ∉ R_{4}

**∴** **R _{4} is not reflexive.**

(ii). **Symmetric:**

If (a, b) ∈ R_{4}, then (b, a) ∈ R_{4}

But (b, a) ∉ R_{4}

[∵ R_{4} = {**(a, b)** (b, c) (c, a)}]

So, ∀ a, b ∈ A, if (a, b) ∈ R_{4}, then (b, a) ∉ R_{4}.

⇒ R_{4} is not symmetric.

It is sufficient to show only one case of ordered pairs violating the definition.

**∴** **R _{4} is not symmetric.**

(iii). **Transitivity:**

We have,

(a, b) ∈ R_{4} and (b, c) ∈ R_{4}

⇒ (a, c) ∈ R_{4}

But, is it so?

No, (a, c) ∉ R_{4}

So, it is enough to determine that R_{4} is not transitive.

∀ a, b, c ∈ A, if (a, b) ∈ R_{4} and (b, c) ∈ R_{4}, then (a, c) ∉ R_{4}.

**∴** **R _{4} is not transitive.**

Rate this question :

Fill in the blanks in each of the

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a^{2} – b^{2}| < 8}. Then R is given by _______.

State True or False for the statements

Every relation which is symmetric and transitive is also reflexive.

Mathematics - ExemplarState True or False for the statements

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

Mathematics - ExemplarState True or False for the statements

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

Mathematics - Exemplar