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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
\( a * b = a – b \quad \forall \, a, b \in \mathbb{Z}^+ \)
Concept:
A binary operation must satisfy the closure property, meaning the result must always belong to the same set.
Solution:
Take \( a = 2 \), \( b = 5 \), where \( a, b \in \mathbb{Z}^+ \).
\( a * b = 2 – 5 = -3 \)
But \( -3 \notin \mathbb{Z}^+ \).
Conclusion:
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 \( \mathbb{Z}^+ \).