Question

The number of reflexive relations on a set \( A \) of \( n \) elements is equal to:

• A. \( 2^{n^2} \)
• B. \( n^2 \)
• C. \( 2^{n(n-1)} \)
• D. \( n^{2-n} \)

Hint:

To count reflexive relations, fix the diagonal pairs (required for reflexivity) and compute the choices for the remaining off-diagonal pairs, which is \( 2^{n(n-1)} \).

Answer 1

The Correct Option is C

Step 1: Reflexive Relation Definition.
A relation \( R \) on a set \( A \) with \( n \) elements is reflexive if every element \( a \in A \) relates to itself: \( (a, a) \in R \) for all \( a \in A \). This requires all diagonal pairs \( (a_1, a_1), (a_2, a_2), \ldots, (a_n, a_n) \) to be in \( R \).
Step 2: Total Possible Pairs.
The total number of ordered pairs \( (x, y) \) where \( x, y \in A \) is \( n \times n = n^2 \). This represents all possible elements in relation \( R \).
Step 3: Reflexive Condition Impact.
Reflexivity mandates that the \( n \) pairs \( (a_i, a_i) \) for \( i = 1 \) to \( n \) are included. This leaves the off-diagonal pairs \( (x, y) \) where \( x \neq y \). The number of such pairs is: \[\nn^2 - n.\n\] Each of these \( n^2 - n \) pairs can either be in \( R \) or not, offering 2 choices (included/excluded) per pair.
Step 4: Counting Reflexive Relations.
There are \( 2^{n^2 - n} \) ways to choose which off-diagonal pairs are included in \( R \). Since diagonal pairs are required, the total number of reflexive relations is: \[\n2^{n^2 - n} = 2^{n(n-1)}.\n\]
Step 5: Result Verification.
For \( n = 1 \), with set \( A = \{a\} \), only \( (a, a) \) is needed, yielding 1 reflexive relation: \( 2^{1(1-1)} = 2^0 = 1 \).
For \( n = 2 \), set \( A = \{a, b\} \), with diagonal pairs \( (a, a), (b, b) \) fixed, and off-diagonal pairs \( (a, b), (b, a) \) giving \( 2^{2(2-1)} = 2^2 = 4 \) relations, confirming the result.