Question:
On the set of integers Z, the operation defined as following is not a binary operation.
On the set of integers Z, the operation defined as following is not a binary operation.
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For a binary operation, "Closure" is key—the result must never leave the original set.
Updated On: Jul 3, 2026
- $a*b = a+b+1, \forall a,b \in Z$
- $a*b = a^b, \forall a,b \in Z$
- $a*b = a+b-ab, \forall a,b \in Z$
- $a*b = lcm(a,b), \forall a,b \in Z$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
A binary operation on a set $S$ must satisfy the closure property: for any $a, b \in S$, $a*b$ must also be in $S$.
Step 2: Meaning
We check if the result of each operation remains an integer for all possible integer inputs $a$ and $b$.
Step 3: Analysis
In option (B), if $b$ is a negative integer (e.g., $a=2, b=-1$), then $2^{-1} = 1/2$, which is not an integer ($1/2 \notin Z$).
Step 4: Conclusion
Since the operation $a^b$ does not always produce an integer when $b < 0$, it is not a binary operation on $Z$. Final Answer: (B)
A binary operation on a set $S$ must satisfy the closure property: for any $a, b \in S$, $a*b$ must also be in $S$.
Step 2: Meaning
We check if the result of each operation remains an integer for all possible integer inputs $a$ and $b$.
Step 3: Analysis
In option (B), if $b$ is a negative integer (e.g., $a=2, b=-1$), then $2^{-1} = 1/2$, which is not an integer ($1/2 \notin Z$).
Step 4: Conclusion
Since the operation $a^b$ does not always produce an integer when $b < 0$, it is not a binary operation on $Z$. Final Answer: (B)
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