Question

Find the total number of distinct binary relations that can be defined over a set \( A \) containing exactly 3 elements.

• A. \( 9 \)
• B. \( 64 \)
• C. \( 512 \)
• D. \( 27 \)

Hint:

Keep these related relation counting formulas memorized for matching questions:

• Total Relations = \( 2^{n^2} \)
• Total Reflexive Relations = \( 2^{n^2 - n} \)
• Total Symmetric Relations = \( 2^{\frac{n(n+1)}{2}} \)

Answer 1

The Correct Option is C

Concept: A binary relation defined on a set \( A \) is any subset of the Cartesian product set \( A \times A \). Therefore, the total number of distinct relations is equal to the total number of subsets in the power set of \( A \times A \), calculated using the formula: \[ \text{Total Relations} = 2^{n(A \times A)} = 2^{n^2} \] Where \( n \) represents the total number of elements in set \( A \).

Step 1: Calculate the size of the Cartesian product set.
The problem states that set \( A \) contains exactly 3 elements (\( n = 3 \)). Find the total number of coordinate pairs in the Cartesian product: \[ n(A \times A) = n \times n = 3 \times 3 = 9 \]

Step 2: Calculate the total number of relation power subsets.
Raise 2 to the power of the Cartesian product size to find the total number of possible subsets: \[ \text{Total Relations} = 2^9 \]

Step 3: Evaluate the final exponential value.
Multiplying out the base-2 value yields: \[ 2^9 = 512 \]