By considering each element of R,determine the relation R' on X with the smallest number of elements satisfying;
(i)R proper subset R'
(ii)R' is symmetric
I think you mean that R' is a proper subset of R, right?(i)R proper subset R'
Cuz, otherwise you have this problem.
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Let X be a finite set. Then R can have at most $\displaystyle n^2$ elements where |X|=n. So you are saying that given any Relation R on X then you can find another relation R' on X that contains R which means that |R'|>|R|. But since R' is a relation on X there exists (according to your hypothesis) another set R'' that containts R' thus, |R''|>|R'|. But then you can use this argument again and again until your relation on X that has cardinality greater than $\displaystyle n^2$, which is impossible.