On Q , the set of all rational numbers a binary operation * is defined by a * b = a + b 2 . Show that * is not associative on Q .

Given binary operation on Q is a * b = a + b 2 for all a , b ∈ Q . * is associative if a * b * c = a * b * c Let a , b , c in Q . Then a * b * c = a * b + c 2 = a + b + c 2 2 ⇒ a * b * c = 2 a + b + c 4 a * b * c = a + b 2 * c = a + b 2 + c 2 = a + b + 2 c 4 Thus, a * b * c ≠ a * b * c Thus, * is not associative on Q .

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b - a b , for all a , b ∈ S . Prove that * is commutative as well as associative.

We have, S = Q - { 1 } and * is defined on S as a * b = a + b - a b , for all a , b ∈ S * is commutative if a * b = b * a Commutativity : Let a , b ∈ S . Then, a * b = a + b - a b = b + a - b a = b * a Therefore a * b = b * a , ∀ a , b ∈ S Thus, * is commutative on S . * is associative if a * b * c = a * b * c Associativity : Let a , b , c ∈ S . Then, a * b * c = a * b + c - b c = a + b + c - b c - a b + c - b c = a + b + c - b c - a b - a c + a b c a * b * c = a + b - a b * c = a + b - a b + c - a + b - a b c = a + b + c - a b - a c - b c + a b c Therefore, a * b * c = a * b * c , ∀ a , b , c ∈ S Thus , * is associative on S . So, * is commutative as well as associative.

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE

R D Sharma Solutions for Chapter: Binary Operations, Exercise 1: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Binary Operations, Exercise 1: EXERCISE

Attempt the free practice questions on Chapter 3: Binary Operations, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-1 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Binary Operations, Exercise 1: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 4

Check the commutativity and associativity of the following binary operation:

$' *'$ on $Q$ defined by $a * b = a b^{2}$ for all $a, b \in Q$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 4

Check the commutativity and associativity of the following binary operation:

$' *'$ on $a * (b * c) \neq (a * b) * c$ defined by $a * b = a + a b$ for all $a, b \in Q$ ?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 4

Check the commutativity and associativity of the following binary operation:

$' *'$ on $N$ , defined by $a * b = a^{b}$ for all $a, b \in N$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 4

Check the commutativity and associativity of the following binary operation:

$' *'$ on $Q$ defined by $a * b = \frac{a b}{4}$ for all $a, b \in Q$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 4

Check the commutativity and associativity of the following binary operation:

$' *'$ on $Z$ defined by $a * b = a + b - a b$ for all $a, b \in Z$ .

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 5

If the binary operation $o$ is defined by $a o b = a + b - a b$ on the set $Q - {- 1}$ of all rational numbers other than $- 1$ , shown that $o$ is commutative on $Q - \{- 1\}$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 13

On $Q$ , the set of all rational numbers a binary operation $*$ is defined by $a * b = \frac{a + b}{2}$ . Show that $*$ is not associative on $Q$ .