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Binary Operation

Is the binary...

Question

Is the binary operation $$ defined on the set of integer $z$ by the rule $a b = a - b + 2$ commutative ?

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Solution

No, as commutative property does not follow subtraction.

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Q. On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.

Q. State the reason for the following Binary Operation *, defined on the set Z of integers, to be not commutative :

a^{*} b = a b^{3}

Q. Which of the following is true?
(a) * defined by

a * b = \frac{a + b}{2}

is a binary operation on Z
(b) * defined by

a * b = \frac{a + b}{2}

is a binary operation on Q
(c) all binary commutative operations are associative
(d) subtraction is a binary operation on N

Q. On Z an operation * is defined by a * b = a² + b² for all a, b ∈ Z. The operation * on Z is
(a) commutative and associative
(b) associative but not commutative
(c) not associative
(d) not a binary operation

Q. Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative.

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