Introduction to Math Proof

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Question No. 1: Let R be the relation on the real numbers given by xRy iff |x-y|≤2. Prove that this relation is Reflexive, Symmetric but not transitive.

Question No. 2: Define a relation ~ on R by, x~y if x-y is rational. Prove that ~ is an equivalence relation.

Question No. 3: Define a relation P that is reflexive but not symmetric and not transitive

Oct 21st, 2017

1. For any real x, xRx because |x - x| = 0 <= 2. The relation is reflexive.

If xRy, then |x - y| <=2. Since, |y - x| = |x - y|, we conclude that yRx and the relation is symmetric.

Set x = 0, y = 2, and z = 4. Then |x - y| = 2 = |y - z|, however, |x - z| = 4 > 2. It means that xRy, yRz but 

x not R z. The relation is not transitive.

2. If x~x, then x - x = 0 is a rational number. The relation is reflexive.

If x~y, then x - y is a rational number and y - x = -(x - y) is a rational number, too. The relation is symmetric.

If x~y and y~z, then both x - y and y - z are rational numbers, x - z = (x - y) + (y - z) is a rational number and x~z. The relation is transitive.

3. Define the following relation R on the set of real numbers. xRy if and only if either x = y or x = 2, y = 1, or x = 1, y = 3. Then the relation R is reflexive. On the other hand, 2R1 is true but 1R2 is not true. Thus, the relation is not symmetric. Finally, 2R1 and 1R3 is true but 2R3 is not true, so the relation is not transitive.

Apr 13th, 2015

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