A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) . Show that * is commutative and associative. Find the identity element for * on A, if any

Solution:

A = N x N and * be the binary operation on A defined by

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

(a, b) * (c, d) ∈ A a, b, c, d ∈ N

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

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

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

Operation * is commutative.

Now, let (a, b), (c, d), (e, f) ∈ A

a, b, c, d, e, f ∈ N

[(a, b) * (c, d )] * (e, f) = (a + c,b + d ) * (e, f) = (a + c + e, b + d + f)

(a, b) * [(c, d ) * (e, f)] = (a, b) * (c + e, d + f) = (a + c + e, b + d + f)

\ [(a, b) * (c, d)] * (e, f) = (a, b) * [(c, d) * (e, f)]

Operation * is associative.

An element e = (e₁, e₂) ∈ A will be an identity element for the operation

* if a + e = a = e * a for all a = (a₁, a₂ ) ∈ A i.e., (a₁ + e₁, a₂ + e₂) = (a₁, a₂) = (e₁ + a₁, e₂ + a₂), which is not true for any element in A.

Therefore, operation * does not have any identity element

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.4 Question 11

A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) . Show that * is commutative and associative. Find the identity element for * on A, if any.

Summary:

A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Hence we conclude that Operation * is commutative. and associative and does not have any identity element