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Question

Which of the following is true?
(a) * defined by a*b=a+b2 is a binary operation on Z
(b) * defined by a*b=a+b2 is a binary operation on Q
(c) all binary commutative operations are associative
(d) subtraction is a binary operation on N

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Solution

(b) * defined by a*b=a+b2 is a binary operation on Q.

Let us check each option one by one.

(a)
If a=1 and b=2,a * b=a+b2 =1+22 =32Z

Hence, (a) is false.

(b)
a * b=a+b2Q, a, bQFor example: Let a=32, b=56Qa*b=32+562 =9+512 =1412 =76Q

Hence, (b) is true.

(c)
Commutativity:
Let a, bN. Then, a * b=2ab =2ba = b * aTherefore, a * b=b * a, a, bN

Thus, * is commutative on N.

Associativity:
Let a, b, cN. Then, a * b * c=a * 2bc =2a*2bca * b * c=2ab * c =2ab*2cTherefore,a * b * ca * b * c

Thus, * is not associative on N.
Therefore, all binary commutative operations are not associative.
Hence, (c) is false.

(d) Subtraction is not a binary operation on N because subtraction of any two natural numbers is not always a natural number.
For example: 2 and 4 are natural numbers.
2-4 = -2 which is not a natural number.
Hence, (d) is false.

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