Home
Class 12
MATHS
Consider the binary operations*: RxxR-...

Consider the binary operations`*: RxxR->R` and `o: RxxR->R` defined as `a*b=|a-b|` and `aob=a` for all `a ,\ b in Rdot` Show that `*` is commutative but not associative, `o` is associative but not commutative.

Text Solution

Verified by Experts

It is given that `*: RxxR ->R` and `o: R xx R->R` defined as `a*b=|a-b|` and `a o b = a , AA a , b in R`.
For `a, b in R` , we have `a*b=|a-b|` and `b*a=|b-a|=|-(a-b)| = |a-b|`
`therefore a*b=b*a`
The `*` operation is commutative.
`(1*2)*3 = (|1-2|)*3 = 1*3 = |1-3|=2`
`1*(2*3) = 1*(|2-3|) = 1*1 = |1-1|=0`
`therefore (1*2)*3 ne 1*(2*3)`
The operation `*` is not associative. ...
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    NCERT ENGLISH|Exercise SOLVED EXAMPLES|50 Videos

  • RELATIONS AND FUNCTIONS

    NCERT ENGLISH|Exercise EXERCISE 1.4|13 Videos

  • PROBABILITY

    NCERT ENGLISH|Exercise EXERCISE 13.2|18 Videos

  • THREE DIMENSIONAL GEOMETRY

    NCERT ENGLISH|Exercise EXERCISE 11.3|14 Videos

Similar Questions

Explore conceptually related problems

Consider the binary operations ** \: R\ xx\ R -> R and o\ : R\ xx\ R -> R defined as a **\ b=|a-b| and aob=a for all a,\ b\ in R . Show that ** is commutative but not associative, o is associative but not commutative.

The binary operation **: RxxR->R is defined as a**b=2a+b . Find (2**3)**4 .

Let * be a binary operation on Z defined by a*b= a+b-4 for all a ,\ b in Zdot Show that * is both commutative and associative.

Consider the binary operation * : R xx R rarr R and o : R xx R rarr R defined as a * b = |a-b| and a o b = a, AA a, b in R . Is o distributive over * ? Justify your answer.

Consider the binary operations ""^('ast') and 'o' on R defined as a^(ast)b=abs(a-b) and aob=a , for all a, b in R . Show that a^(ast)(boc)=(a^(ast)b)oc for all a, b, c in R.

If * is a binary operation in N defined as a*b =a^(3)+b^(3) , then which of the following is true : (i) * is associative as well as commutative. (ii) * is commutative but not associative (iii) * is associative but not commutative (iv) * is neither associative not commutative.

Let * be a binary operation on Q-{-1} defined by a * b=a+b+a b 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}

On Z an operation * is defined by a*b= a^2+b^2 for all a ,\ b in Z . The operation * on Z is (a)commutative and associative (b)associative but not commutative (c) not associative (d) not a binary operation

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=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.