Question

Determine whether or not each of the definitions given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.
On ${Z^ + }$, define $*$ by $a*b = \left| {a - b} \right|$

Answer 1

Here we will firstly write the given range of the integers according to the given conditions. Then we will take the operation equation and assume a case when the value of $a$ is equal to the value of $b$ and check whether the operation is a binary operation or not.

Complete Step by Step Solution:
It is given the range of the integers will be ${Z^ + }$, which states that the range of the integers is equal to all the positive integers.
Now we will form the condition of the expression given in the question. Therefore, the given operation $*$ is
$\Rightarrow a*b = \left| {a - b} \right|$
Now we will assume the case when the value of $a$ is equal to the value of $b$ i.e. if $a = b$.
Then the value of the operation $\left| {a - b} \right|$ is equal to zero i.e. $\left| {a - b} \right| = 0$.
It is given that the value of the range of the integers is always possible and zero doesn’t belong to the range given i.e. $0 \notin {Z^ + }$ which states that the operation $*$ is not a binary operation.

Hence, on ${Z^ + }$, define $*$ by $a*b = \left| {a - b} \right|$, operation given is not a binary operation.

Note:
Here, we need to know the way of writing an equation in the general form of a variable with its variable range. Generally, the value of the variable is positive natural numbers or integers.
We should also know that integers are the numbers which can be positive or negative but integers can never be in fractional form or decimal form.
Natural numbers are the positive integer numbers. Even numbers are any integer that can be divided exactly by 2 is an even number. The last digit is 0, 2, 4, 6 or 8.
Binary operations are the general operations like addition or subtraction or other basic arithmetic operations.