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Determine whether the operation is a binary operation or not
Given: An operation \( * \) on \( \mathbb{N} \) defined by
\( a * b = a + b – 2 \quad \forall \, a, b \in \mathbb{N} \)
Concept:
A binary operation on a set must satisfy the closure property, meaning the result of the operation on any two elements must also belong to the same set.
Solution:
Let \( a = 1 \) and \( b = 1 \), where \( a, b \in \mathbb{N} \).
\( a * b = 1 + 1 – 2 = 0 \)
But \( 0 \notin \mathbb{N} \) (natural numbers do not include 0 in this context).
Conclusion:
Since the result is not always a natural number, the set is not closed under this operation.
❌ Therefore, the operation is NOT a binary operation on \( \mathbb{N} \).