Smallest Equivalence Relation on Real Numbers: Proving with Line y-x=1

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SUMMARY

The smallest equivalence relation S on the set of real numbers R that contains all points on the line y - x = 1 is defined by the set of all pairs (x, y) such that x - y is a constant. This relation satisfies the properties of reflexivity, symmetry, and transitivity as required for equivalence relations. Specifically, the equivalence classes can be represented as lines parallel to y - x = 1, effectively grouping real numbers based on their distance from this line.

PREREQUISITES
  • Understanding of equivalence relations in set theory
  • Familiarity with the properties of reflexivity, symmetry, and transitivity
  • Basic knowledge of Cartesian coordinates and linear equations
  • Ability to visualize geometric representations of mathematical concepts
NEXT STEPS
  • Study the properties of equivalence relations in more depth
  • Explore examples of equivalence relations in different mathematical contexts
  • Learn about equivalence classes and their applications
  • Investigate the geometric interpretation of relations in the Cartesian plane
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Mathematicians, educators, and students studying set theory and equivalence relations, as well as anyone interested in the geometric interpretation of mathematical concepts.

dharper8861
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1) Recall that an equivalence relation S on set R ( R being the reals) is a subset of R x R such that

(a) For every x belonging to R (x,x) belongs to S
(b) If (x,y) belongs to S, then (y,x) belongs to S
(c) If (x,y) belongs to S and (y,z) belongs to S then (x,z) belongs to S

What is the smallest equivalence relation S on the Set R of real numbers that contains all the points in the line y - x = 1. Prove your answer.

Can anyone help figure this out? I am pretty lost on this one.
 
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Have you tried drawing a picture of S? You know it has at least the line y - x = 1. What does (a) tell you in terms of your picture? And (b)? Does (c) work for your picture?
 

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