Verify the associative binary operation for ( Z + , * ) where a * b = 2 a b for all a , b ∈ Z + and also find whether it is commutative or not.

Given that, * is defined on Z + . a * b = 2 a b Checking associative property. i.e., ( a * b ) * c = a * ( b * c ) i.e., ( 2 a b ) * c = a * ( 2 b c ) i.e., 2 2 a b c = 2 a 2 b c i.e., 4 a b c = 4 a b c ∴ Given binary operation on a * b = 2 a b is associative on Z + . Checking commutative property on Z + , i.e., a * b = b * a i.e., 2 a b = 2 b a ∴ Given binary operation a * b = 2 a b is commutative on Z + .

Verify the associative binary operation for ( Q , * ) where a * b = a b + 3 for all a , b ∈ Q and also find whether it is commutative or not.

Given that, * is defined on Q . a * b = a b + 3 Checking associative property. i.e., ( a * b ) * c = a * ( b * c ) i.e., ( a b + 3 ) * c = a * ( b c + 3 ) i.e., ( a b + 3 ) c + 3 = a ( b c + 3 ) + 3 i.e., a b c + 3 c + 3 = a b c + 3 a + 3 For values a ≠ c , above statement is false. ∴ Given binary operation on a * b = a b + 3 is not associative on Q . Checking commutative property on Q , i.e., a * b = b * a i.e., a b + 3 = b a + 3 ∴ Given binary operation a * b = a b + 3 is commutative on Q .

Verify the associative binary operation for ( * ) where a * b = max ( a , b ) for all a , b ∈ R and also find whether it is commutative or not.

Given that, * is defined on R . a * b = max a , b Checking associative property, i.e., ( a * b ) * c = a * ( b * c ) i.e., max ( a , b ) , c = max a , ( b , c ) Example: m a x 1 , 2 , 3 = m a x 3 , 2 , 1 = 3 ∴ Given binary operation a * b = a , b is associative on R . Checking commutative property, i.e., a * b = b * a i.e., max ( a , b ) = max ( b , a ) ∴ Given binary operation, a * b = max a , b is commutative on R .

Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 3: EXERCISE

Author:Parthasarathi Mukhopadhyay & Manabendra Nath Mukherjee

Parthasarathi Mukhopadhyay Mathematics Solutions for Exercise - Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 3: EXERCISE

Attempt the practice questions from Exercise 3: EXERCISE with hints and solutions to strengthen your understanding. RUDIMENTS of MATHEMATICS For Class 12 of +2 Level solutions are prepared by Experienced Embibe Experts.

Questions from Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 3: EXERCISE with Hints & Solutions

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12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 1

Verify the associative binary operation for $(Z^{+}, *)$ where $a * b = 2 a b$ for all $a, b \in Z^{+}$ and also find whether it is commutative or not.

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 1

Verify the associative binary operation for $(Q, *)$ where $a * b = a b + 3$ for all $a, b \in Q$ and also find whether it is commutative or not.

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 1

Verify the associative binary operation for $(*)$ where $a * b = \max (a, b)$ for all $a, b \in R$ and also find whether it is commutative or not.

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 2