# Thread: Total no. of reflexive relations on a set of n elements

1. ## Total no. of reflexive relations on a set of n elements

The number of reflexive relations on an n-element set is 2n2−n
Can anybody tell me why so?

2. ## Re: Total no. of reflexive relations on a set of n elements

Originally Posted by nikhilbhr
The number of reflexive relations on an n-element set is 2n2−n Can anybody tell me why so?
Hint Every relation $\displaystyle R$ on $\displaystyle A=\{a_1,\ldots,a_n\}$ is an element of $\displaystyle \mathcal{P}( A\times A)$ . So, we have $\displaystyle \textrm{card}\;\mathcal{P}( A\times A)=2^{\;\textrm{card }\mathcal{P}( A\times A)}}=2^{n^2}$ relations on $\displaystyle A$ . Could you continue?

3. ## Re: Total no. of reflexive relations on a set of n elements

Originally Posted by nikhilbhr
The number of reflexive relations on an n-element set is $\displaystyle 2^{n^2-n}$ Can anybody tell me why so?[/I]
Here is a different approach.
Any reflexive relation on a set of n elements must contain the diagonal. The diagonal has n elements.
So how many off-diagonal elements are there?
Now look at your answer, $\displaystyle 2^{n^2-n}$.
How are they related?

4. ## Re: Total no. of reflexive relations on a set of n elements

Originally Posted by Plato
Here is a different approach.
Any reflexive relation on a set of n elements must contain the diagonal. The diagonal has n elements.
So how many off-diagonal elements are there?
Now look at your answer, $\displaystyle 2^{n^2-n}$.
How are they related?
I understood your approach but can u tell me how you are correlating off-diagonal elements with reflexive relation.

5. ## Re: Total no. of reflexive relations on a set of n elements

Originally Posted by nikhilbhr
I understood your approach but can u tell me how you are correlating off-diagonal elements with reflexive relation.
How many subsets of off-diagonal elements are there?

6. ## Re: Total no. of reflexive relations on a set of n elements

total elements in that matrix=nXn
total diagonal element=n
so obviously power set off-diagonal elements=2^(n^2-1).
But i have problem with the approach u are using .Can please how off-diagonal elements are correlated with reflexive relation..
Thanx for help.

7. ## Re: Total no. of reflexive relations on a set of n elements

Denote $\displaystyle \Delta$ the diagonal of $\displaystyle A\times A$ with $\displaystyle A=\{a_1,\ldots,a_n\}$ . Every reflexive relation $\displaystyle R$ on $\displaystyle A$ has the form $\displaystyle R=\Delta\cup B$ with $\displaystyle B\subset A\times A-\Delta$ . But $\displaystyle A\times A-\Delta$ has $\displaystyle n^2-n$ elements, so there are $\displaystyle 2^{n^2-n}$ reflexive relations on $\displaystyle A$ .

8. ## Re: Total no. of reflexive relations on a set of n elements

Thank u very much for your valuable help..Now i got where was the problem.Actually R= (set of diagonal element) U B ,i havent thought of this expression

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# number of reflexive relations on an n-element set

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