Q2 of 107 Page 24

For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

On R – {–1}, define a ∗ b =

It is given that On R, * – {– 1}, define a ∗ b =

ϵ R for b ≠ -1, so the operation * is binary.


We can see that 1 * 2 = and 2 * 1=


⇒ 1 * 2 ≠ 2 * 1; where 1,2 ϵ R – {-1}


⇒ the operation * is not commutative.


Now, we can observed that


(1 * 2) * 3 =


1 * (2 * 3) = 1 *


⇒ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1,2,3 ϵ R * – {– 1}


⇒ The operation * is not associative.


More from this chapter

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2

For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

On Z+, define a ∗ b = 2ab

 2

For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

On Z+, define a ∗ b = ab

 3

consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}. Write the operation table of the operation ∧.

 4

Consider a binary operation ∗ on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).

(i) Compute (2 ∗ 3) ∗ 4 and 2 ∗ (3 ∗ 4)


(ii) Is ∗ commutative?


(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).


(Hint: use the following table)


Table 1.2