Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this. On Z + , defined * by a*b = |a - b| Here, Z + denotes the set of all non-negative integers.

Binary Operation on Non-Negative Integers

Determine whether the operation is a binary operation or not

Given: The set \( \mathbb{Z}^+ = \{0,1,2,3,\dots\} \) and operation \( * \) defined by

\( a * b = |a – b| \quad \forall \, a, b \in \mathbb{Z}^+ \)

A binary operation must satisfy the closure property.

Let \( a, b \in \mathbb{Z}^+ \).

\( a * b = |a – b| \)

The absolute difference of two non-negative integers is always a non-negative integer.

\( |a – b| \geq 0 \Rightarrow |a – b| \in \mathbb{Z}^+ \)

The set is closed under this operation.

✔ Therefore, the operation is a binary operation on \( \mathbb{Z}^+ \).

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