Let a binary operation '*' be defined on a set A. The operation will be commutative if
Answer Details
A binary operation '*' on a set A is said to be commutative if the order in which the elements are combined does not affect the result. In other words, for any two elements a and b in A, the operation a * b is equal to b * a.
Let's say we have a set A with elements {a, b, c}. For the operation '*' to be commutative, it should hold true for all possible combinations of elements in A.
1. We need to check if a * b is equal to b * a. If a * b = b * a for all a and b in A, then the operation is commutative. 2. We also need to check if (a * b) * c is equal to a * (b * c). If this equation holds true for all a, b, and c in A, then the operation is associative, but it does not necessarily guarantee commutativity. 3. Finally, we need to check if (b ο c) * a is equal to (b * a) ο (c * a). If this equation holds true for all a, b, and c in A, then the operation is commutative.
In summary: - If the operation a * b = b * a holds true for all a and b in A, then the operation is commutative. - If (a * b) * c = a * (b * c) holds true for all a, b, and c in A, the operation is associative, but not necessarily commutative. - If (b ο c) * a = (b * a) ο (c * a) holds true for all a, b, and c in A, the operation is commutative.
Therefore, the option that satisfies the condition of commutativity is **a * b = b * a**.