Class 12th

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:

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

Therefore, the operation* is associative.

An element $e=\left(e_1,e_2\right)$ will be an identity element for the operation* if

$a*e=a=e*a\forall a=\left(a_1,a_2\right)\in A$

$i.e.,\left(a_1+e_1,a_2+e_2\right)=\left(a_1,a_2\right)=\left(e_1+a_1,e_2+a_2\right)$ 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.

(i) On Q the operation* is defined as a*b=a-b.

It can be observed that:

$\begin{array}{l}\frac{1}{2}.\frac{1}{3}=\frac{1}{2}-\frac{1}{3}=\frac{3-2}{6}=\frac{1}{6}"and"\frac{1}{3}.\frac{1}{2}=\frac{1}{3}-\frac{1}{2}=\frac{2-3}{6}=\frac{-1}{6}\\ \therefore \frac{1}{2}.\frac{1}{3}\ne \frac{1}{3}.\frac{1}{2}where,\frac{1}{2},\frac{1}{3}\in Q\end{array}$

Thus, the operation* is not commutative.

It can also be observed that:

$\begin{array}{l}\left(\frac{1}{2}.\frac{1}{3}\right).\frac{1}{4}=\left(\frac{1}{2}-\frac{1}{3}\right).\frac{1}{4}=\frac{1}{6}.\frac{1}{4}=\frac{1}{6}-\frac{1}{4}=\frac{2-3}{12}=\frac{-1}{12}\\ \frac{1}{2}.\left(\frac{1}{3}.\frac{1}{4}\right)=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{4}\right)=\frac{1}{2}.\frac{1}{12}=\frac{1}{2}-\frac{1}{12}=\frac{6-1}{12}=\frac{5}{12}\\ \therefore \left(\frac{1}{2}.\frac{1}{3}\right).\frac{1}{4}\ne \frac{1}{2}.\left(\frac{1}{3}.\frac{1}{4}\right)where,\frac{1}{2},\frac{1}{3},\frac{1}{4}\in Q\end{array}$

Thus, the operation* is not associative.

(ii) On Q the operation* is defined as a*b=a²+b.

For $a,b\in Q$ , we have:

$\begin{array}{l}a.b=a^2+b^2=b^2+a^2=b.a\\ \therefore a*b=b*a\end{array}$

Thus, the operation* is commutative.

It can be observed that:

$\begin{array}{l}\left(1*2\right)*3=\left(12+22\right)*3=\left(1+4\right)*3=5*3=52+32=25+9=34\\ 1*\left(2*3\right)=1*\left(22+32\right)=1*\left(4+9\right)=1*13=12+132=1+169=170\\ \therefore \left(1*2\right)*3\ne 1*\left(2*3\right),where,1,2,3\in Q\end{array}$

Thus, the operation* is not commutative.

(iii) On Q the operation* is defined as a*b=a+ab.

It can be observed that:

$\begin{array}{l}1.2=1+1*2=1+2=3\\ 2.1=2+2*1=2+2=4\\ \therefore 1.2\ne 2.1,where,1,2\in Q\end{array}$

Thus, the operation* is not commutative.

It can also be observed that:

$\begin{array}{l}\left(1.2\right).3=\left(1+1*2\right).3=3.3=3+3*3=3+9=12\\ 1.\left(2.3\right)=1.\left(2+2*3\right)=1+1*8=9\\ \therefore \left(1.2\right).3\ne 1.\left(2.3\right),where,1,2,3\in Q\end{array}$

Thus, the operation* is not associative.

(iv) On Q the operation* is defined as a*b=(a-b)²

For $a,b\in Q$ , we have:

$\begin{array}{l}a*b={\left(a-b\right)}^2\\ b*a={\left(b-a\right)}^2={\left[-\left(a-b\right)\right]}^2={\left(a-b\right)}^2\\ \therefore a*b=b*a\end{array}$

Thus, the operation* is commutative.

It can be observed that:

$\begin{array}{l}\left(1.2\right).3={\left(1.2\right)}^2.3={\left(-1\right)}^2.3=1.3={\left(1.3\right)}^2={\left(-2\right)}^2=4\\ 1.\left(2.3\right)=1.{\left(2.3\right)}^2=1.{\left(-1\right)}^2=1.1={\left(1.1\right)}^2=0\\ \therefore \left(1.2\right).3\ne 1.\left(2.3\right),where,1,2,3\in Q\end{array}$

Thus, the operation* is not associative.

(v) On Q the operation* is defined as a.b=ab/4

For $a,b\in Q$ , we have:

$\begin{array}{l}a.b=\frac{ab}{4}=b\frac{a}{4}=b.a\\ \therefore a*b=b*a\end{array}$

Thus, the operation* is commutative.

For $a,b,c\in Q$ , we have:

$\begin{array}{l}\left(a.b\right).c=\frac{ab}{4}.c=\frac{\frac{ab}{4}.c}{4}=\frac{abc}{16}\\ a.\left(b.c\right)=a.b\frac{c}{4}=\frac{a.\frac{bc}{4}}{4}=\frac{abc}{16}\\ =\therefore \left(a*b\right)*c\ne a*\left(b*c\right)\end{array}$

Thus, the operation* is associative.

(vi) On Q the operation* is defined as a*b=ab²

It can be observed that:

$\begin{array}{l}\frac{1}{2}.\frac{1}{3}=\frac{1}{2}.{\left(\frac{1}{3}\right)}^2=\frac{1}{2}.\frac{1}{9}=\frac{1}{18}\\ \frac{1}{3}.\frac{1}{2}=\frac{1}{3}.{\left(\frac{1}{2}\right)}^2=\frac{1}{3}.\frac{1}{4}=\frac{1}{12}\\ \therefore \frac{1}{2}.\frac{1}{3}\ne \frac{1}{3}.\frac{1}{2},where,\frac{1}{2},\frac{1}{3}\in Q\end{array}$

Thus, the operation* is not commutative.

It can be observed that:

$\begin{array}{l}\left(\frac{1}{2}.\frac{1}{3}\right).\frac{1}{4}=\left[\frac{1}{2}.{\left(\frac{1}{3}\right)}^2\right].\frac{1}{4}=\frac{1}{18}.\frac{1}{4}=\frac{1}{18}.{\left(\frac{1}{4}\right)}^2=\frac{1}{18*16}\\ \frac{1}{2}.\left(\frac{1}{3}.\frac{1}{4}\right)=\frac{1}{2}.\left[\frac{1}{3}.{\left(\frac{1}{4}\right)}^2\right]=\frac{1}{2}.\frac{1}{48}=\frac{1}{2}.{\left(\frac{1}{48}\right)}^2=\frac{1}{2*{\left(48\right)}^2}\\ \therefore \left(\frac{1}{2}.\frac{1}{3}\right).\frac{1}{4}\ne \frac{1}{2}.\left(\frac{1}{3}.\frac{1}{4}\right),where,\frac{1}{2},\frac{1}{3},1.4\in Q\end{array}$

Thus, the operation* is not associative.

Hence, the operations defined (ii), (iv), (v) are commutative and the operation defined in (v) is associative.

The binary operation * on N is defined as a * b = L.C.M. of a and b.

(i) 5 * 7 = L.C.M. of 5 and 7 = 35

20 * 16 = L.C.M of 20 and 16 = 80

(ii) It is known that: 

L.C.M of a and b = L.C.M of b and a & mnForE; a, b ∈ N. 

∴a * b = b * a

Thus, the operation * is commutative.

(iii) For a, b, c ∈ N, we have:

(a * b) * c = (L.C.M of a and b) * c = LCM of a, b, and c

a * (b * c) = a * (LCM of b and c) = L.C.M of a, b, and c

∴ (a * b) * c = a * (b * c) 

Thus, the operation * is associative.

(iv) It is known that:

L.C.M. of a and 1 = a = L.C.M. 1 and a &mnForE; a ∈ N 

⇒ a * 1 = a = 1 * a &mnForE; a ∈ N

Thus, 1 is the identity of * in N.

(v) An element a in N is invertible with respect to the operation * if there exists an element b in N, such that a * b = e = b * a.

Here, e = 1

This means that: 

L.C.M of a and b = 1 = L.C.M of b and a

This case is possible only when a and b are equal to 1.

Thus, 1 is the only invertible element of N with respect to the operation *.