Home
Class 12
MATHS
Let A" "=" "NxxN and * be the binary ...

Let `A" "=" "NxxN` 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.

Text Solution

Verified by Experts

In `A = NxxN`
(a,b)*(c,d) = (a+ c,b+d)
Let a,b,c,`in`,N
and (a,b)* {(c,d)* (e,f)}
`= (a,b)**(c+e,d+f)`
`= {a+(c+e),b+(d+f)}`
`{(a+c)+e,(b+d)+f)}`
`=(a+c,b+d)**(e,f)`
`{(a,b)** (c,d)}**(e,f)`
`therefore` Opeation * is associative .
Let `(e,f)in N xxN`
and `(a,b)**(e,f) = (a,b)=(e,f)**(a,b)`
`rArr (a+e,b+f) = (a,b)= (e+a,f+b)`
`rArr " " e= 0, f= 0`
Which are not the elements of N.
`therefore` In `A = N xx N`, the identity elements does not exist
With respect to (a,b) * (c,d) = (a+c,b+d)
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|19 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN|Exercise Exercise 1.3|14 Videos

  • PROBABIILITY

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|19 Videos

  • THREE-DIMENSIONAL GEOMETRY

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|23 Videos

Similar Questions

Explore conceptually related problems

Let A=NN 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.

Let A = N xx N and ** be the binary opertation on A defined by (a, b) ** (c, d) = (a+c, b+d) . Show that ** is commutative and associative.

Let A=RR 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.

Let A=NxNa n d^(prime)*' be a binaryoperation on A defined by (a , b)*(C , d)=(a c , b d) for all a , b , c , d , in Ndot Show that '*' is commutative and associative binary operation on A.

Let A=NxxN , and let * be a binary operation on A defined by (a , b)*(c , d)=(a d+b c , b d) for all (a , b), (c , d) in NxxNdot Show that: * is commutative on Adot (ii) * is associative on Adot

Let A=NxN , and let * be a binary operation on A defined by (a , b)*(c , d)=(a d+b c , b d) for all (a , b),c , d) in NxNdot Show that : '*' is commutative on A '*^(prime) is associative onA A has no identity element.

Let R_(0) denote the set of all non-zero real numbers and let A=R_(0)xx R_(0) .If * is a binary operation on A defined by (a,b)*(c,d)=(ac,bd) for all (a,b),(c,d)in A* show that * is both commutative and associative on A( ii) Find the identity element in A

Let * be a binary operation on Q-{-1} defined by a*b=a+b+ab for all a,b in Q-{-1}. Then,Show that * is both commutative and associative on Q-{-1} (ii) Find the identity element in Q-{-1}

Let A=N uu{0}xx N uu{0} and let * be a binary operation on A defined by (a,b)*(c,d)=(a+c,b+d) for all (a,b),(c,d)in A. Show that * is commutative on A.

Let '**' be a binary operation defined on NxxN by : (a, b)**(c, d)=(a+c,b+d) . Prove that '**' is commutative and associative.