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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

Apr 13th, 2015
2.Reflexive:  x-x = 0 is rational, so x~x

  Symmetric:  x-y = p/q, then y-x = -p/q

  so x~y ===> y~x.

  Transitive:  If x-y = p/q and y-z = r/s, then

  x-z = p/q+r/s which is still rational.

  So x~y and y~z implies x~z.


  • Reflexive and transitive: The relation ≤ on p Or any preorder
  • Symmetric and transitive: The relation P on N, defined as aPbab ≠ 0.
  • Reflexive and symmetric: The relation P on Z, defined as aPb ↔ "ab is divisible by at least one of 2 or 3

Apr 14th, 2015

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