Let a binary operation '*' be defined on a set A. The operation will be commutative if

Analysis

The question asks for the definition of a commutative binary operation on a set $A$. In mathematics, binary operations have specific properties that define how elements interact with one another.

Concept Explanation

A binary operation $*$ on a set $A$ is a rule that assigns to each ordered pair of elements $(a, b)$ from $A$ a unique element $a * b$ in $A$.

• Commutative Property: An operation is commutative if the order of the elements does not change the result. That is, for any two elements $a$ and $b$ in the set, $a * b$ must be equal to $b * a$.
• Associative Property: This is represented by the expression $(a * b) * c = a * (b * c)$. This property deals with the grouping of elements, not the order.
• Distributive Property: This involves two different operations (e.g., $*$ and $\circ$) and describes how one operation interacts with another over a set.

Step-by-Step Solution

1. Examine Option A ($a * b = b * a$): This is the standard definition of commutativity. If you swap the positions of $a$ and $b$, the result remains the same.
2. Examine Option B ($(a * b) * c = a * (b * c)$): This is the definition of the associative property, not the commutative property.
3. Examine Option C ($(b \circ c) * a = (b * a) \circ (c * a)$): This represents the distributive property of the operation $*$ over the operation $\circ$.
4. Conclusion: Since Option A matches the mathematical definition of a commutative operation, it is the correct choice.

Exam Tip

To remember the difference:

• Commutative: Think of "Commuting" (traveling to work and back). The distance is the same regardless of the direction. Order changes, result stays.
• Associative: Think of "Associating" (grouping friends). It doesn't matter which two friends you group together first; the total group remains the same. Grouping changes, result stays.