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

Solution:

It is given that

* : R × R → R and o : R × R → R defined as

a * b = a - b and aob = a, "a, b ∈ R

For a, b ∈ R,

we have a * b = |a - b| and b * a = |b - a|

= |- (a - b)| = |a - b|

⇒ a * b = b * a

⇒ The operation * is commutative.

(1 * 2) * 3 = (|1 - 2|) * 3 = 1 * 3 = |1 - 3| = 2

1* (2 * 3) = 1 * (|2 - 3|) = 1 * 1 = |1 - 1| = 0

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3)

where1, 2, 3 ∈ R

⇒ The operation * is not associative.

Now, consider the operation o :

It can be observed that 1o2 = 1and 2o1 = 2.

⇒ 1o2 ≠ 2o1 (where 1, 2 ∈ R)

⇒ The operation o is not commutative.

Let a, b, c ∈ R. Then, we have:

(aob) oc = aoc = a ao (boc) = aob = a

⇒ (aob) oc = ao (boc)

⇒ The operation o is associative.

Now, let a, b, c ∈ R, then we have:

a * (boc) = a * b = |a - b|

(a * b)o (a * c)

= (|a - b|)o (|a - c|) = |a - b|

Hence,

a * (boc) = (a * b)o (a * c)

Now,

1o (2 * 3) = 1o (|2 - 3|) = 1o1 = 1

(1o2) * (1o3) = 1 * 1 = |1 - 1| = 0

⇒ 1o (2 * 3) ≠ (1o2) * (1o3) where 1, 2, 3 ∈ R

⇒ The operation o does not distribute over*

---

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 12

Consider the binary operations *: R × R → R and o: R × R → R defined as a * b = a - b and aob = 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 * (boc) = (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

Summary:

The operation * is commutative. The operation * is not associative. The operation o is not commutative. The operation o is associative. The operation o does not distribute over*