The following relations are defined on the set of real numbers :
aRb if 1 + ab > 0
Find whether these relations are reflexive, symmetric or transitive.
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 1 + ab > 0
Check for Reflexivity:
For a ∈ ℝ
If aRa,
⇒ 1 + aa > 0
⇒ 1 + a2 > 0
If a is a real number.
[Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers. Real numbers are so called because they are ‘Real’ not ‘imaginary’.]
This means, even if ‘a’ was negative.
a2 = positive.
a2 + 1 = positive
And any positive number is greater than 0.
Hence, 1 + a2 > 0
⇒ aRa is true.
So, ∀ a ∈ ℝ, then aRa is true.
⇒ R is reflexive.
∴ R is reflexive.
Check for Symmetry:
∀ a, b ∈ ℝ
If aRb,
⇒ 1 + ab > 0
Replace a by b and b by a, we get
⇒ 1 + ba > 0
Whether we write ab or ba, it is equal.
ab = ba
So, 1 + ba > 0
⇒ bRa is true.
So, if aRb is true, then bRa is true.
∀ a, b ∈ ℝ
⇒ R is symmetric.
∴ R is symmetric.
Check for Transitivity:
∀ a, b, c ∈ ℝ
If aRb and bRc.
⇒ 1 + ab > 0 and 1 + bc > 0
⇒ 1 + ac > 0 or not.
Let us check.
1 + ab > 0 means ab > –1.
1 + bc > 0 means bc > –1.
1 + ac > 0 means ac > –1.
If ab > –1 and bc > –1.
⇒ ac > –1 should be true.
Take a = –1, b = 0.9 and c = 1
ab > –1
⇒ (–1)(0.9) > –1
⇒ –0.9 > –1, is true on the number line.
bc > –1
⇒ (0.9)(1) > –1
⇒ 0.9 > –1, is true on the number line.
ac > –1
⇒ (–1)(1) > –1
⇒ –1 > –1, is not true as –1 cannot be greater than itself.
⇒ ac > –1 is not true.
⇒ 1 + ac > 0 is not true.
⇒ aRc is not true.
So, if aRb is true and bRc is true, then aRc is not true.
∀ a, b, c ∈ ℝ
⇒ R is not transitive.
∴ R is not transitive.