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Relations and Functions Ncert Solutions Maths class 12th Maths Class 12th Relations and Functions

**53. Let 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.**

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Vishal Baghel | Contributor-Level 10

5 months ago

A = N × N

* is a binary operation on A and is defined by:

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

Let (a, b), (c, d) ∈ A

Then, a, b, c, d ∈ N

We have:

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

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

[Addition is commutative in the set of natural numbers]

∴(a, b) * (c, d) = (c, d) * (a, b)

Therefore, the operation * is commutative.

Now, let (a, b), (c, d), (e, f) ∈A

Then, a, b, c, d, e, f ∈ N

We have:

$\begin{array}{l} ((a, b) . (c, d)) . (e, f) = (a + c, b + d) . (e, f) = (a + c + e, b + d + f) \\ (a, b) . ((c, d) . (e, f)) = (a, b) . (c + e, d + f) = (a + c + e, b + d + f) \\ ∴ ((a, b) . (c, d)) . (e, f) = (a, b) . ((c, d) . (e, f)) \end{array}$

Therefore, the operation* is associative.

An element $e = (e_{1}, e_{2})$ will be an identity element for the operation* if

$a * e = a = e * a \forall a = (a_{1}, a_{2}) \in A$

$i . e ., (a_{1} + e_{1}, a_{2} + e_{2}) = (a_{1}, a_{2}) = (e_{1} + a_{1}, e_{2} + a_{2})$ which is not true for any element in A.

Therefore, the operation* doe

...more

A = N × N

* is a binary operation on A and is defined by:

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

Let (a, b), (c, d) ∈ A

Then, a, b, c, d ∈ N

We have:

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

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

[Addition is commutative in the set of natural numbers]

∴(a, b) * (c, d) = (c, d) * (a, b)

Therefore, the operation * is commutative.

Now, let (a, b), (c, d), (e, f) ∈A

Then, a, b, c, d, e, f ∈ N

We have:

Therefore, the operation* is associative.

An element $e = (e_{1}, e_{2})$ will be an identity element for the operation* if

$a * e = a = e * a \forall a = (a_{1}, a_{2}) \in A$

$i . e ., (a_{1} + e_{1}, a_{2} + e_{2}) = (a_{1}, a_{2}) = (e_{1} + a_{1}, e_{2} + a_{2})$ which is not true for any element in A.

Therefore, the operation* does not have any identity element.

Which is not true for any element in A.

Therefore, the operation * does not have any identity element.

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**53. Let 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.**

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53. Let 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.

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**53. Let 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.**