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

In summary, an equivalence relation on the set of real numbers is a subset of the Cartesian product of real numbers that satisfies three conditions: (a) every element in R is related to itself, (b) if two elements are related, then their order can be reversed, and (c) if two elements are related and one is also related to a third element, then the first and third elements are also related.To find the smallest equivalence relation that contains all points on the line y - x = 1, we can draw a picture of S and see that it must include the line. Condition (a) tells us that every point on the line is related to itself, and condition (b) tells us that if two points are
  • #1
dharper8861
1
0
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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  • #2
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?
 

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

What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between two elements of a set. It is a binary relation that is reflexive, symmetric, and transitive.

How is an equivalence relation different from other relations?

An equivalence relation is different from other relations because it has three specific properties: reflexivity, symmetry, and transitivity. These properties must hold for all elements in the set for the relation to be considered an equivalence relation.

What is an example of an equivalence relation?

An example of an equivalence relation is the relation "is equal to" on the set of real numbers. This relation is reflexive, symmetric, and transitive, as any number is equal to itself, two numbers are equal to each other if and only if they have the same value, and if two numbers are equal to each other and a third number is equal to the second, then the third number is also equal to the first.

Why are equivalence relations important?

Equivalence relations are important in mathematics because they allow us to classify objects into different equivalence classes. This helps us to better understand and organize different mathematical concepts and structures, and can also be useful in solving problems and proving theorems.

How are equivalence relations used in other fields?

Equivalence relations are not only used in mathematics, but also in other fields such as computer science, linguistics, and philosophy. In computer science, equivalence relations are used in data structures and algorithms. In linguistics, they are used to study language and categorize different linguistic forms. In philosophy, they are used to define the concept of identity and distinguish between different objects or concepts.

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