Let $A = Z \times Z$ and * be a binary operation on $A$ defined by $(a, b) * (c, d) = (a d + b c, b d)$ .
Find the identity element for * in the set $A$ .

Open in App

Solution

Let $A = Z \times Z$

i) identity element for $$

A defined by

$(a, b) (c, d) = (a d + b c, b d)$

Let (e, f) be identity

$(a, b) * (e, f) = (a f + b e, b f)$

so $(a f + b e, b f) = (a, b)$

$b f = b | a f + b e = a$

so $f = 1 | a (1) + b e = a$

so $e = 0$ .

identity $(e, f) = (0, 1)$

$(0, 1)$ is the identity element

Suggest Corrections

Similar questions

Let S = Z x (Z - {0}) and the binary operation * is defined as (a , b) * (c ,d) = (ad + bc , bd) for all a,b,c,d ϵZ. The identity element of of S for the binary operation * on S is __

Q. Let be a binary operation on z defined by

a \times b = a + b - 4 \forall a, b \in z

. Find the identity element

let * be a binary operation on z defined by a*b=a+b-4, for all a,b&z, find the identity element

Q. Let

A = Q \times Q

and let

*

be a binary operation on

A

defined by

(a, b) * (c, d) = (a c, b + a d)

for

(a, b) (c, d) ϵ A

. Determine, whether

*

is commutative and associative. Then, with respect to

*

A

(i)

Find the identity element in

A

(i i)

Find the invertible elements of

A

Q. Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z
(i) Show that '*' is both commutative and associative.
(ii) Find the identity element in Z.
(iii) Find the invertible elements in Z.

Join BYJU'S Learning Program

Let A=Z×Z and * be a binary operation on A defined by (a,b)∗(c,d)=(ad+bc,bd).Find the identity element for * in the set A.

Let A=Z×Zi) identity element for ∗A defined by(a,b)∗(c,d)=(ad+bc,bd)Let (e, f) be identity(a,b)∗(e,f)=(af+be,bf)so (af+be,bf)=(a,b)bf=b|af+be=aso f=1|a(1)+be=aso e=0.identity (e,f)=(0,1)(0,1) is the identity element

Let $A = Z \times Z$ and * be a binary operation on $A$ defined by $(a, b) * (c, d) = (a d + b c, b d)$ .
Find the identity element for * in the set $A$ .