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How mny relations they satisfy the property of equivalence relation as  well as partial order over set A={1,2,3,4}.

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Equivalence relation: reflexive, symmetric, and transitive
Partial order relation: reflexive, antisymmetric, and transitive

|A| = n
|AxA| = $n^2$

we have total $n^2$ ordered pairs, for n diagonal pairs we don't have any choice, we have to keep them to satisfy reflexive property.
Now (n^2-n) ordered pairs are only left. 
if (a,b) is present in our relation, then Equivalence relation says (b,a) should also be present as it needs to satisfy symmetric property, but partial order relation oppose it because it has to satisfy antisymmetric property.

Hence, there will only be one such relation which is both Equivalence and partial order.

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