Determine whether or not each of the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation, give justification for this.
(iv) On $Z^{+}$ , defined $a * b = | a - b |$

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Solution

On $Z^{+}$ , defined $a * b = | a - b |$
It is seen that for each a,b, $\in Z^{+}$ , there is a unique element $| a - b | i n Z^{+} .$
This means that $$ carries each pair (a,b)to a unique element $a b = | a - b | i n Z^{+}$ . Therefore, $*$ is a binary operation.

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Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this. (iv) On Z+, defined a∗b=|a−b|

On Z+, defined a∗b=|a−b| It is seen that for each a,b, ∈Z+, there is a unique element |a−b| in Z+. This means that ∗ carries each pair (a,b)to a unique element a∗b=|a−b| in Z+. Therefore, ∗ is a binary operation.

Determine whether or not each of the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation, give justification for this.
(iv) On $Z^{+}$ , defined $a * b = | a - b |$

On $Z^{+}$ , defined $a * b = | a - b |$
It is seen that for each a,b, $\in Z^{+}$ , there is a unique element $| a - b | i n Z^{+} .$
This means that $$ carries each pair (a,b)to a unique element $a b = | a - b | i n Z^{+}$ . Therefore, $*$ is a binary operation.