NCERT Solutions
Class 12 Maths
Relations and Functions
Misc. Ex. Q12
Consider the binary operations *: R × R → R and o: R × R → R defined as a ∗ b = |a − b| and a o b = a, ∀ a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a*(b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
It is given that *: R × R → and o: R × R → R is defined as a ∗ b = |a − b| and a o b = a, ∀ a, b ∈ R
For a, b ∈ R, we have
a ∗ b = |a − b| and b ∗ a = |b − a| = |−(a − b)| = |a − b|
So, a * b = b * a
Hence, the operation * is commutative.
It can be observed that
(1 ∗ 2) ∗ 3 = (|1 − 2|) ∗ 3 = 1 ∗ 3 = |1 − 3| = 2 and
1 ∗ (2 ∗ 3) = 1 ∗ (|2 − 3|) = 1 ∗ 1 = |1 − 1| = 0
So, (1 ∗ 2) ∗ 3 ≠ 1 ∗ (2 ∗ 3) where 1, 2, 3 ∈ R.
Hence, the operation * is not associative.
Now, consider the operation o
It can be observed that 1 o 2 = 1 and 2 o 1 = 2
So, 1 o 2 ≠ 2 o 1 where 1, 2 ∈ R.
Hence, the operation o is not commutative.
Let a, b, c ∈ R. Then, we have
(a o b) o c = a o c = a and
a o (b o c) = a o b = a
So, a o b) o c = a o (b o c), where a, b, c ∈ R
Hence, the operation o is associative.
Now, let a, b, c ∈ R, then we have
a * (b o c) = a * b =|a − b|
(a * b) o (a * c) = (|a − b|)o(|a − c|) = |a − b|
Hence, a * (b o c) = (a * b) o (a * c).
Now,
1 o (2 ∗ 3) = 1o(|2 − 3|) = 1o1 = 1
(1 o 2) * (1 o 3) = 1 * 1 = |1 − 1| = 0
So, 1 o (2 * 3) ≠ (1 o 2) * (1 o 3) where 1, 2, 3 ∈ R
Hence, the operation o does not distribute over *.