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Consider the binary operations **:RRxxRR...

Consider the binary operations `**:RRxxRRtoRRand@:RRxxRRtoRR` defined as `a**b=|a-b|anda@b=a` for all `a,binRR` then-

A

`**` is commutative but not associative on `RR`

B

`@` is associative on `RR`.

C

`@` is not distribution over `**`

D

`@` is commutative on `RR`

Text Solution

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The correct Answer is:
A,B,C
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Let ** and @ be two binary operations on RR defined as, a**b=|a-b|anda@b=a for all a,binRR . Examine the commutativity and associativity of ** and @ on RR . Show also that ** is distributative over @ but @ is not distributive over ** .

Prove that the binary operation ** on RR defined by a**b=a+b+ab for all a,binRR is commutative and associative.

Knowledge Check

  • If the binary oprations ** on RR is defined by a**b=a+b+ab for all a,binRR where on R.H.S. we have usual addition, subtraction and multiplication of real numbers. The relation ** is---

    A
    not commutative
    B
    associative
    C
    commutative
    D
    not associative

  • Consider a binary opertion ** on N defined as a **b=a ^(3) +b ^(3). Choose the correct answer.

    A
    Is `**`both associative and commutative ?
    B
    Is `**` commutative but not associative ?
    C
    Is `**` associative but not commutative ?
    D
    Is `**` neither comutative nor associative ?

    • Similar Questions

    Explore conceptually related problems

    Show that identity of the binary operation ** on RR defined by a**b=|a+b| for all a,binRR, does not exist.

    Prove that the identity element of the binary opeartion ** on RR defined by a**b = min. (a,b) for all a,binRR , does not exist.

    Find the identity element of the binary operation ** on ZZ defined by a**b=a+b+1 for all a,binZZ .

    On the set QQ^(+) of all positive rational numbers if the binary operation ** is defined by a**b=(1)/(4)ab for all a,binQQ^(+) , find the identity element in QQ^(+) . Also prove that any element in QQ^(+) is invertible.

    Discuss the commutativity and associativity ** on RR defined by a**b=|ab| for all a,binRR .

    On the set Q^+ of all positive rational numbers if the binary operation * is defined by a * b = 1/4ab for all a,b in Q^+ find the identity element in Q^+ .

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