Q. 5 A

# The following relations are defined on the set of real numbers :

aRb if a – b > 0

Find whether these relations are reflexive, symmetric or transitive.

Answer :

Let set of real numbers be ℝ.

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.

We have

aRb if a – b > 0

**Check for Reflexivity:**

For a ∈ ℝ

If aRa,

⇒ a – a > 0

⇒ 0 > 0

But 0 > 0 is not possible.

Hence, aRa is not true.

So, ∀ a ∈ ℝ, then aRa is not true.

⇒ R is not reflexive.

**∴** **R is not reflexive.**

**Check for Symmetry:**

∀ a, b ∈ ℝ

If aRb,

⇒ a – b > 0

Replace a by b and b by a, we get

⇒ b – a > 0

[Take a = 12 and b = 6.

a – b > 0

⇒ 12 – 6 > 0

⇒ 6 > 0, which is a true statement.

Now, b – a > 0

⇒ 6 – 12 > 0

⇒ –6 > 0, which is not a true statement as –6 is not greater than 0.]

⇒ bRa is not true.

So, if aRb is true, then bRa is not true.

∀ a, b ∈ ℝ

⇒ R is not symmetric.

**∴** **R is not symmetric.**

**Check for Transitivity:**

∀ a, b, c ∈ ℝ

If aRb and bRc.

⇒ a – b > 0 and b – c > 0

⇒ a – c > 0 or not.

Let us check.

a – b > 0 means a > b.

b – c > 0 means b > c.

a – c > 0 means a > c.

If a > b and b > c,

⇒ a > b, b > c

⇒ a > b > c

⇒ a > c

Hence, aRc is true.

So, if aRb is true and bRc is true, then aRc is true.

∀ a, b, c ∈ ℝ

⇒ R is transitive.

**∴** **R is 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