MCQ
Consider a binary operation $∗$ on $N$ defined as $a ∗ b = a^3 + b^3.$
- A$∗$ is both associative and commutative.
- ✓$∗$ is commutative but not associative.
- C$∗$ is neither commutative nor associative.
- D$∗$ is associative but not commutative.
✓
Answer
Correct option: B.
$∗$ is commutative but not associative.
Given that the binary operation $∗$ on $N$ is defined as $a∗b = a^3 + b^3$.
Apply the given binary operation on $ b∗a$.
$b∗a = b^3 + a^3 = a^3 + b^3$
It shows that the value of $a∗b$ is equal to that of $b∗a$.
So, the operation is commutative.
Consider different values of the variable as $a = 1, b = 2$ and $c = 3.$
Apply the given binary operation on $(a∗b)∗c$.
$(a∗b)∗c = (1∗2)∗3 = (1^3 + 2^3)∗3 = 9^3 + 3^3 = 729 + 27 = 756$
Apply the given binary operation on $a∗(b∗c).$
$(a∗b)∗c = 1∗(2∗3) = 1∗(2^3 + 3^3) = 1^3 + 35^3 = 42876$
$(a∗b)∗c \neq a∗(b∗c)$
So the operation is not associative.
Therefore, the given operation is commutative but not associative.
Apply the given binary operation on $ b∗a$.
$b∗a = b^3 + a^3 = a^3 + b^3$
It shows that the value of $a∗b$ is equal to that of $b∗a$.
So, the operation is commutative.
Consider different values of the variable as $a = 1, b = 2$ and $c = 3.$
Apply the given binary operation on $(a∗b)∗c$.
$(a∗b)∗c = (1∗2)∗3 = (1^3 + 2^3)∗3 = 9^3 + 3^3 = 729 + 27 = 756$
Apply the given binary operation on $a∗(b∗c).$
$(a∗b)∗c = 1∗(2∗3) = 1∗(2^3 + 3^3) = 1^3 + 35^3 = 42876$
$(a∗b)∗c \neq a∗(b∗c)$
So the operation is not associative.
Therefore, the given operation is commutative but not associative.
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