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how many reflexive relation on A={1,2,3,4,5},could be definde

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Let set A = {1 , 2}. In reflexive relation on A \times A are such that (a,b) \in A \times A \implies (b,a) \in A .

Hence, the following are the reflexive relation on A\times A :

\{(1, 1), (2, 2)\} \\ \{(1, 1), (2, 2), (1, 2)\} \\ \{(1, 1), (2, 2), (1, 2), (2, 1)\} \\ \{(1, 1), (2, 2), (2, 1)\} .

A relation has ordered pairs (a,b). Now a can be chosen in n ways and same for b. So set of ordered pairs contains n2 pairs. Now for a reflexive relation, (a,a) must be present in these ordered pairs. And there will be total n pairs of (a,a), so number of ordered pairs will be n2-n pairs. So total number of reflexive relations is equal to 2n(n-1).

Formulas of the number of reflexive relations on an n-element set is 2^{n^2-n} .

Hence, the number of reflexive relation on A={1,2,3,4,5} are 2^{5^2-5} = 2^{25-5}=2^{20} = 1048576.

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