Proving R is an Equivalence Relation: Steps and Explanation

In summary, the conversation discusses how to prove that a relation R on X is an equivalence relation. It is given that R satisfies the properties of reflexivity (for all a in X, (a,a) is in R) and transitivity (for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R). In order for R to be an equivalence relation, it must also satisfy the property of symmetry (for a,b in X, if (a,b) is in R, then (b,a) is also in R). The conversation suggests considering specific examples, such as (a,b) and (b,b) or (b,c) and (c,c
  • #1
ragnes
4
0
1. Let R be a relation on X that satisfies
a) for all a in X, (a,a) is in R
b) for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R.
Show that R is an equivalence relation.



2. In order for R to be an equivalence relation, the following must be true:
1) for all a in X, (a,a) is in R
2) for a,b in X, if (a,b) is in R, then (b,a) is also in R
3) for a,b,c in X, if (a,b) and (b,c) is in R, then (a,c) is in R.



3. The first part is given by the definition of an equivalence relation. I'm stuck on proving part b. Help please!
 
Physics news on Phys.org
  • #2
how about considering (a,b) and (b,b) for the second part?
 
  • #3
lanedance said:
how about considering (a,b) and (b,b) for the second part?

Proof by contradiction, using lanedance's example.

Or consider (b,c) and (c,c).
 

What are equivalence relations?

Equivalence relations are a mathematical concept that describes the relationship between two objects that are considered equivalent. These relations are reflexive, symmetric, and transitive, meaning that they are defined in such a way that every object is equivalent to itself, two objects are equivalent if and only if they are equivalent to each other, and if one object is equivalent to another and that object is equivalent to a third, then the first and third objects are also equivalent.

What are some examples of equivalence relations?

Some examples of equivalence relations include equality of numbers, congruence of geometric figures, and similarity of shapes. In each of these cases, the relation satisfies the three properties of reflexivity, symmetry, and transitivity.

How are equivalence relations represented?

Equivalence relations can be represented in various ways, such as by using a symbol or notation to show the relationship between two objects. For example, the equal sign (=) is used to represent the equivalence relation of equality between numbers. In set theory, the symbol ≅ is used to represent equivalence between sets.

What is the importance of equivalence relations?

Equivalence relations are important in mathematics as they provide a way to classify objects into distinct groups based on their relationships with one another. They also allow for the simplification and generalization of mathematical concepts, making problem-solving and understanding abstract ideas easier.

How are equivalence relations used in real-world applications?

Equivalence relations have various real-world applications, such as in computer science and data analysis. In computer science, equivalence relations are used to categorize data and structures, while in data analysis, they are used to identify patterns and similarities among data sets. In everyday life, equivalence relations can also be applied in fields such as linguistics, where they are used to classify languages and determine their relationships with one another.

Similar threads

On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
708
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
513
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
1
Views
458
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
2
Views
262
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
510
Prove ##a + b = b + a## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
808
Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
260
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
573
Prove ##a\cdot b = b \cdot a ##using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
684

Hot Threads

Recent Insights

Back
Top