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21 votes

What is the possible number of reflexive relations on a set of 5 elements?

- $2^{10}$
- $2^{15}$
- $2^{20}$
- $2^{25}$

34 votes

Best answer

A relation consists of set of ordered pairs $(a,b)$. Here, $a$ can be chosen in $n$ ways and similarly, $b$ can be chosen in $n$ ways. So, totally $n^2$ possible ordered pairs are possible for a relation. Now each of these ordered pair can either be present in the relation or not- 2 possibilities for each of the $n^2$ pair. So, total number of possible relations = $$2^{\left(n^2\right)}$$

Now, for a relation $R$ to be reflexive, ordered pairs $\left\{(a,a) \mid a \in S \right\}$, must be present in $R$. i.e.; the relation set $R$ must have $n$ ordered pairs fixed. So, number of ordered pairs possible is $ n^2 - n$ and hence total number of reflexive relations is equal to $$ 2^{\left(n^2-n\right)}$$

for $n=5$, answer will be, $2^{5^2-5}=2^{20}$

Therefore,** option C** is correct

0 votes

The number of reflexive relations is given by $2^{n^{2}-n}$. The reasoning is not always very clear so here is the explanation:

- If the relation must be
*reflexive*means we need to have all the self-pairs, e.g. (1,1) for all elements in the set. So choice for these is one as they**have to be included, no matter what.** - Now come the other pairs left.
**These may or may not be included and it will not affect the overall reflexivity of the relation, as all self-pairs are already selected.**

So, total self-pairs = $n$, and remaining pairs = ${n^2}-n$ , which is total minus the self-pairs and the left pairs still have a choice.

Hence total number of reflexive pairs = $2^{n^{2}-n}$

Substituting $n$=5 for this question we get $2^{20}$, which is **option (C).**