Question

Let * be a binary operation defined on N×N by (a, b)*(c, d)=(a+c, b+d).Find the identity element for * if it exists.​

Answer 1

Answer :

(0,0)

Solution :

Given Binary Operation :

* : N×N → N×N , (a,b)*(c,d) = (a+c , b+d)

★ Identify element :-

Let (e,e) be the identity element for the given binary operation * .

Thus ,

=> (a,b)*(e,e) = (a,b)

=> (a + e , b + e) = (a,b)

Comparing the ordered pairs in LHS and and RHS , we have ;

=> a + e = a

=> e = a - a

=> e = 0

Also ,

=> b + e = b

=> e = b - b

=> e = 0

→ We obtained an unique value of e .

Thus ,

The identity element for the binary operation * exists which is equal to (0,0) .

Answer 2

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Check the binary operation * is commutative :

We know that, * is commutative if (a, b) * (c, d) = (c, d) * (a, b) ∀ a, b, c, d ∈ R

L.H.S =(a, b) * (c, d)

=(a + c, b + d)

R. H. S = (c, d) * (a, b)

=(a + c, b + d)

Hence, L.H.S = R. H. S

Since (a, b) * (c, d) = (c, d) * (a, b) ∀ a, b, c, d ∈ R

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

Check the binary operation * is associative :

We know that * is associative if (a, b) * ( (c, d) * (x, y) ) = ((a, b) * (c, d)) * (x, y) ∀ a, b, c, d, x, y ∈ R

L.H.S = (a, b) * ( (c, d) * (x, y) ) = (a+c+x, b+d+y)

R.H.S = ((a, b) * (c, d)) * (x, y) = (a+c+x, b+d+y)

Thus, L.H.S = R.H.S

Since (a, b) * ( (c, d) * (x, y) ) = ((a, b) * (c, d)) * (x, y) ∀ a, b, c, d, x, y ∈ R

Thus, the binary operation * is associative

Checking for Identity Element:

e is identity of * if (a, b) * e = e * (a, b) = (a, b)

where e = (x, y)

Thus, (a, b) * (x, y) = (x, y) * (a, b) = (a, b) (a + x, b + y)

= (x + a , b + y) = (a, b)

Now, (a + x, b + y) = (a, b)

Now comparing these, we get:

a+x = a

x = a -a = 0

Next compare: b +y = b

y = b-b = 0

Since A = N x N, where x and y are the natural numbers. But in this case, x and y is not a natural number. Thus, the identity element does not exist.

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