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Determine whether or not each of the def...

Determine whether or not each of the defination of `**` given below gives a binary opertion. In the even that `**` is not a binary opertion, give justification for this.
(i) On `Z ^(+),` define `**` by `a ** b =a -b`
(ii) On `Z ^(+),` define `**` by` a **b =ab`
(iii) On R, define `**` by `a **b =ab ^(2)`
(iv) On `Z ^(+),` define `**` by `a ** b = |a-b|`
(v) On `Z ^(+),` define `**` by `a ** b =a`

Text Solution

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The correct Answer is:
(i) NO (ii) Yes (iii) es (iv) Yes (v) Yes
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For each opertion ** difined below, determine whether ** isw binary, commutative or associative. (i) On Z, define a **b =a - b (ii) On Q, define a **b =ab +1 (iii) On Q, define a **b = (ab)/( 2) (iv) On Z ^(+), define a **b = 2 ^(ab) (v) On Z ^(+), define a **b=a ^(b) (vi) On R - {-1}, define a **b= (a)/( b +1)

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Knowledge Check

  • Number of binary opertions on the set {a,b} are

    A
    10
    B
    16
    C
    20
    D
    8

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