First slide

Combinations

Question

Let $X$ be a set containing $n$ elements. The number of reflexive relations that can be defined on $X$ is

Moderate

A

$2^{n^{2}}$
B

$2^{n^{2} - n}$
C

$n (2^{n^{2} - n})$
D

$n (2^{n^{2}})$

Solution

Let $X = \{x_{1}, x_{2}, \dots x_{n}\}$ The set $X \times X$ contains $n^{2}$ elements.
A reflexive relation $R$ on $X$ must contain $(x_{i}, x_{i})$ for all $1 \leq i \leq n,$
i.e.,R must contain the n elements $(x_{1}, x_{1}), (x_{2}, x_{2})$ $\dots (x_{0}, x_{0})$ and any subset of the set containing remaining $n^{2} - n$ elements. Therefore, the number of reflexive relations that can be defined on $X$ is $2^{n^{2} - n}$

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