How Do I Define an Equivalence Relation on a Subset?

In summary, an equivalence relation is a subset of the cartesian product of two sets, where the ordered pairs in the relation satisfy the properties of transitive, symmetric, and reflexive. In the given example of I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}, the equivalence relation would be a subset of I X I, where the ordered pairs satisfy the given conditions. The concept of a cartesian product is important in understanding equivalence relations in set theory.
  • #1
mcfc
17
0
If I have a subset, how do I define an equivalence relation.
I understand it has to satisfy three properties:transitive, symmetric and reflexive, but I'm not sure how to give an explicit definition of the equivalence relation, for example on I where
[itex]I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}[/itex]
 
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  • #2


Do you know what a cartesian product is? If you don't its a very important topic for anyone learning set theory to know.

If you do, then an equivalence relation R from A to B is a subset of A X B. In other words an equivalence relation R contains those ordered pairs (a,b) [tex]\in[/tex] A X B such that a is related to b by R.

In your example that equivalence relation is a subset of [tex]\Re[/tex] X [tex]\Re[/tex] consisting of those (x,y) [tex]\in[/tex] [tex]\Re[/tex] X [tex]\Re[/tex] such that 0 [tex]\leq[/tex] x [tex]\leq[/tex] 1, 0 [tex]\leq[/tex] y [tex]\leq[/tex] 1.

Hope that makes sense to you.
 
  • #3


CharmedQuark said:
Do you know what a cartesian product is? If you don't its a very important topic for anyone learning set theory to know.

If you do, then an equivalence relation R from A to B is a subset of A X B. In other words an equivalence relation R contains those ordered pairs (a,b) [tex]\in[/tex] A X B such that a is related to b by R.

In your example that equivalence relation is a subset of [tex]\Re[/tex] X [tex]\Re[/tex] consisting of those (x,y) [tex]\in[/tex] [tex]\Re[/tex] X [tex]\Re[/tex] such that 0 [tex]\leq[/tex] x [tex]\leq[/tex] 1, 0 [tex]\leq[/tex] y [tex]\leq[/tex] 1.

Hope that makes sense to you.

HI

That does makes sense, but I can't see how to define an explicit equivalence relation...?
 
  • #4


mcfc said:
...for example on I where
[itex]I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}[/itex]

I x I has the required properties, right?
 
  • #5


bpet said:
I x I has the required properties, right?

sorry...I don't follow(again)
 
  • #6


mcfc said:
sorry...I don't follow(again)

The equivalence relation you gave is a relation on the set I. I X I is the cartesian product of I with itself. Since the relation R is from I to I it is a subset of I X I. An equivalence relation is a set and can be written as such.

Perhaps if you rephrased your question I could be of more help?
 

FAQ: How Do I Define an Equivalence Relation on a Subset?

What is an equivalence relation?

An equivalence relation is a mathematical concept that describes a relationship between two objects or elements. It is a type of binary relation that indicates that two objects are equivalent in some way.

What are the properties of an equivalence relation?

An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element is related to itself. Symmetry means that if element A is related to element B, then element B is also related to element A. Transitivity means that if element A is related to element B and element B is related to element C, then element A is also related to element C.

How is an equivalence relation different from other types of relations?

An equivalence relation is different from other types of relations because it is a special type of relation that satisfies the three properties mentioned above. Other types of relations, such as partial orders or total orders, may not satisfy all three properties.

What are some examples of equivalence relations?

Some common examples of equivalence relations include equality of numbers, congruence of geometric figures, and similarity of shapes. In everyday life, equivalence relations can also be seen in relationships between people, such as siblings or spouses, where certain characteristics or qualities are shared.

How is an equivalence relation used in mathematics?

Equivalence relations are used in mathematics to establish a sense of equivalence or equality between objects. They are often used in proof techniques, such as proof by induction or proof by contradiction, to show that certain objects or elements are equivalent in some way. Equivalence relations are also important in defining mathematical structures, such as groups or equivalence classes.

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