The given relations on R ore mu equivalence relations. For each relation, state

Question

The given relations on R ore mu equivalence relations. For each relation, state which of the properties-reflexive, symmetric, and transitive-fail. Give counterexamples for the properties that a relation doesn't satisfy and proofs for the properties that a relation satisfies. Define xAy iff xy > 0. Define xBy iff xy ge 0. Define xCy iff x + y = 0. Define xDy iff y - x > 2. Define xEy iff |y - x| le 1.

Explanation / Answer

a) Not reflexive because 0*0 is not > 0 Symmetric because xy = yx Transitive because if xy > 0, yz > 0, then x, y, z have the same sign, so xz > 0 b) Not transitive because -1*0 >= 0, 0*1 >= 0, but -1*1 is not >= 0 Symmetric because xy = yx Transitive because if xy >=0, yz >= 0, then x,y,z have the same sign (or at least one of them equals 0), so xz >= 0 c) Not reflexive because 1 + 1 is not 0 Not transitive because -1 + 1 = 0, 1 + -1 = 0, but -1 -1 = -2 is not 0 Symmetric because x + y = y + x d) Not reflexive because 0 - 0 is not > 2 Not symmetric because 6 - 3 > 2, but 3 - 6 is not > 2 Transitive because a - b > 2, b - c > 2 => a - c > 4 > 2 (just adding the first two together) e) Not transitive because |3 - 2|

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