Let * be a binary operation on Z defined by a * b = a + b − 4 for all a , b ∈ Z . Show that ' * ' is both commutative and associative.

Given binary operation on Z is a * b = a + b - 4 for all a , b ∈ Z * is commutative if a * b = b * a Commutativity: Let, a , b ∈ Z . Then, a * b = a + b - 4 = b + a - 4 = b * a Therefore, a * b = b * a , ∀ a , b ∈ Z Thus, * is commutative on Z . * is associative if a * b * c = a * b * c Associativity: Let a , b , c ∈ Z . Then, a * b * c = a * b + c - 4 = a + b + c - 4 - 4 = a + b + c - 8 a * b * c = a + b - 4 * c = a + b - 4 + c - 4 = a + b + c - 8 Therefore, a * b * c = a * b * c , ∀ a , b , c ∈ Z Thus, * is associative on Z .

Let * be a binary operation on Z defined by a * b = a + b − 4 for all a , b ∈ Z . Find the identity element in Z .

Given binary operation on Z is a * b = a + b - 4 for all a , b ∈ Z . Let e be the identity element in Z with respect to * such that a * e = a = e * a , ∀ a ∈ Z a * e = a and e * a = a , ∀ a ∈ Z a + e - 4 = a and e + a - 4 = a , ∀ a ∈ Z e = 4 , ∀ a ∈ Z Thus, 4 is the identity element in Z with respect to * .

Let * be a binary operation on Z defined by a * b = a + b − 4 for all a , b ∈ Z . Find the invertible elements in Z .

Given binary operation on Z is a * b = a + b - 4 for all a , b ∈ Z . Let e be the identity element in Z with respect to * such that a * e = a = e * a , ∀ a ∈ Z a * e = a and e * a = a , ∀ a ∈ Z a + e - 4 = a and e + a - 4 = a , ∀ a ∈ Z e = 4 , ∀ a ∈ Z Thus, 4 is the identity element in Z with respect to * . Let a ∈ Z and b ∈ Z be the inverse of a . Then, a * b = e = b * a a * b = e and b * a = e We know that the value for the e for the given is 4 . a + b - 4 = 4 and b + a - 4 = 4 b = 8 - a ∈ Z Thus, 8 - a is the inverse of a ∈ Z .

Let * be a binary operation on Q 0 (set of non-zero rational numbers) defined by a * b = 3 a b 5 for all a , b ∈ Q 0 . Show that * is commutative as well as associative. Also, find its identity element if it exists.

Given binary operation on Q 0 is a * b = 3 a b 5 for all a , b ∈ Q 0 * is commutative if a * b = b * a Commutativity: Let a , b ∈ Q 0 a * b = 3 a b 5 = 3 b a 5 = b * a Therefore, a * b = b * a , ∀ a , b ∈ Q 0 * is associative if a * b * c = a * b * c Associativity: Let a , b , c ∈ Q 0 a * b * c = a * 3 b c 5 = a 3 b c 5 5 = 3 a b c 25 a * b * c = 3 a b 5 * c = 3 a b 5 c 5 = a b c 25 Therefore, a * b * c = a * b * c , ∀ a , b , c ∈ Q 0 Thus, * is associative on Q 0 . Finding identity element : Let e be the identity element in Q 0 with respect to * such that a * e = a = e * a , ∀ a ∈ Q 0 a * e = a and e * a = a , ∀ a ∈ Q 0 ⇒ a e 5 = a and e a 5 = a , ∀ a ∈ Q 0 ⇒ e = 5 , ∀ a ∈ Q 0 ∵ a ≠ 0 Thus, 5 is the identity element in Q 0 with respect to * .

Let * be a binary operation on Q − { − 1 } defined by a * b = a + b + a b for all a , b ∈ Q − { − 1 } . Show that ' * ' is both commutative and associative on Q − { − 1 } .

Given binary operation on Q - - 1 is a * b = a + b + a b for all a , b ∈ Q - - 1 . * is commutative if a * b = b * a Commutativity: Let a , b ∈ Q - - 1 . Then, a * b = a + b + a b = b + a + b a = b * a Therefore, a * b = b * a , ∀ a , b ∈ Q - - 1 Thus, * is commutative on Q - { - 1 } * is associative if a * b * c = a * b * c Associativity: Let a , b , c ∈ Q - - 1 . 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 ∈ Q - - 1 . Thus, * is associative on Q - { - 1 }

Let * be a binary operation on Q − { − 1 } defined by a * b = a + b + a b for all a , b ∈ Q − { − 1 } . Find the identity element in Q − { − 1 } .

Given binary operation on Q - - 1 is a * b = a + b + a b for all a , b ∈ Q - - 1 . Let e be the identity element in Q - { - 1 } with respect to * such that a * e = a = e * a , ∀ a ∈ Q - - 1 ⇒ a * e = a and e * a = a , ∀ a ∈ Q - - 1 ⇒ a + e + a e = a and e + a + e a = a , ∀ a ∈ Q - - 1 ⇒ e 1 + a = 0 , ∀ a ∈ Q - - 1 ⇒ e = 0 , ∀ a ∈ Q - - 1 ∵ a ≠ - 1 Thus, 0 is the identity element in Q - { - 1 } with respect to * .

Let * be a binary operation on Q − { − 1 } defined by a * b = a + b + a b for all a , b ∈ Q − { − 1 } . Show that every element of Q − { − 1 } is invertible. Also, find the inverse of an arbitrary element.

Given binary operation on Q - - 1 is a * b = a + b + a b for all a , b ∈ Q - - 1 . Let e be the identity element in Q - { - 1 } with respect to * such that a * e = a = e * a , ∀ a ∈ Q - - 1 ⇒ a * e = a and e * a = a , ∀ a ∈ Q - - 1 ⇒ a + e + a e = a and e + a + e a = a , ∀ a ∈ Q - - 1 ⇒ e 1 + a = 0 , ∀ a ∈ Q - - 1 ⇒ e = 0 , ∀ a ∈ Q - - 1 ∵ a ≠ - 1 Thus, 0 is the identity element in Q - { - 1 } with respect to * . Let a ∈ Q - - 1 and b ∈ Q - - 1 be the inverse of a . Then, a * b = e = b * a ⇒ a * b = e and b * a = e We know that the value of e is 0 . ⇒ a + b + a b = 0 and b + a + b a = 0 ⇒ b 1 + a = - a ∈ Q - - 1 ⇒ b = - a 1 + a ∈ Q - - 1 ∵ a ≠ - 1 Thus, - a 1 + a is the inverse of a ∈ Q - - 1 .

E M B I B E

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

EASY

12th CBSE

IMPORTANT

Earn 100

If the binary operation $$ on the set $Z$* is defined by $a * b = a + b - 5$ , the find the identity element with respect to $*$ .

Important Questions on Binary Operations

EASY

12th CBSE

IMPORTANT

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

On the set $Z$ of integers, if the binary operation

*

is defined by

a * b = a + b + 2

, then find the identity element.

MEDIUM

12th CBSE

IMPORTANT

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

Let

*

be a binary operation on $Z$ defined by

a * b = a + b - 4

for all

a, b \in Z

. Show that

' *'

is both commutative and associative.

EASY

12th CBSE

IMPORTANT

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

Let $*$ be a binary operation on $Z$ defined by $a * b = a + b - 4$ for all $a, b \in Z$ .

Find the identity element in $Z$ .

MEDIUM

12th CBSE

IMPORTANT

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

Let $*$ be a binary operation on $Z$ defined by $a * b = a + b - 4$ for all $a, b \in Z$ .

Find the invertible elements in $Z$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 3 - Binary Operations>EXERCISE>Q 2

Let

*

be a binary operation on

Q_{0}

(set of non-zero rational numbers) defined by

a * b = \frac{3 a b}{5}

for all

a, b \in Q_{0}

. Show that

*

is commutative as well as associative. Also, find its identity element if it exists.

MEDIUM

12th CBSE

IMPORTANT

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

Let $*$ be a binary operation on $Q - {- 1}$ defined by $a * b = a + b + a b$ for all $a, b \in Q - {- 1}$ . Show that $' *'$ is both commutative and associative on $Q - {- 1}$ .