State the reason for the following binary operation *, defined on the

1.	State the reason for the following binary operation , defined on the set Z of integers, to be not commutative.a b = ab3
Answer» Let a, b ∈ Z, where Z is set of integers. Given binary operation is a ∗ b = ab³ . Now, b ∗ a = ba³ ≠ ab³ = a ∗ b. ( \(\because\) ba³ ≠ ab³ , where a and b are arbitrary integers ) Therefore, b ∗ a ≠ a ∗ b. Therefore, the binary operation ‘∗’ is not commutative.

State the reason for the following binary operation , defined on the set Z of integers, to be not commutative.a b = ab3

Answer»

Let a, b ∈ Z, where Z is set of integers.

Given binary operation is a ∗ b = ab³ .

Now, b ∗ a = ba³ ≠ ab³ = a ∗ b. ( \(\because\) ba³ ≠ ab³ , where a and b are arbitrary integers )

Therefore, b ∗ a ≠ a ∗ b.

Therefore, the binary operation ‘∗’ is not commutative.

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State the reason for the following binary operation *, defined on the set Z of integers, to be not commutative.a * b = ab3

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State the reason for the following binary operation , defined on the set Z of integers, to be not commutative.a b = ab3