Given a Relation (set of ordered pairs), prove transitivity without going through each pair?

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Give a relation, R, on the set of integers, such as R = {(1,2)(2,2) ... } is there a way to determine transitivity without going through each ordered pair (x,y)(y,z) to see if (x,z) is there?

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No. you have to check for each pair of ordered pair. Sorry no shortcuts