Determine whether or not 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 \( \mathbb{Z}^+ = \{0,1,2,3,\dots\} \) (non-negative integers) and operation \( * \) defined by

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

A binary operation must satisfy the closure property, meaning the result must always belong to the same set.

Take \( a = 2 \), \( b = 5 \), where \( a, b \in \mathbb{Z}^+ \).

\( a * b = 2 – 5 = -3 \)

But \( -3 \notin \mathbb{Z}^+ \).

Since the result is not always a non-negative integer, the set is not closed under this operation.

❌ Therefore, the operation is NOT a binary operation on \( \mathbb{Z}^+ \).

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