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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 27th, 2015

1. for every x , |x-x| = 0 <2

for every x, y |x-y| = |y-x| thus if |x-y|<=2 so does |y-x|

take x= 1, y = 2 z= 3.5 then |2-1|<=2, |3.5-2|<=2 but |3.5-1|>2

2. let x= p/q, y = s/t then x-y = (pt-sq)/qt, y-x = - (pt-sq)/qt are  rational and x-x=0 = 0/1 is rational. Also let z=u/v then x-y = (pt-sq)/qt, y-z = (sv-tu)/tv => x-z=(pt-sq)/qt + (sv-tu)/tv=(pv-uq)/vq

3. x>y+1

Apr 27th, 2015

Correction to 3.

x<|y|+1,

obviously a<|a|+1 for all a.

take x,y=(1,3) shows not symmetric

x,y,z = (2,-2, 1) show not transitive

Apr 27th, 2015

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