Math Problem Statement
Let R be an equivalence relation on a set A . Prove b € [a] = = [a] = [b]
Solution
Let be an equivalence relation on a set , which means that satisfies three properties:
- Reflexivity: for all .
- Symmetry: If , then .
- Transitivity: If and , then .
Let represent the equivalence class of , i.e.,
Proof that :
(1) Assume :
This means that , by the definition of the equivalence class. We want to show that .
- Take any . By definition, . Since is symmetric, we know implies .
- Since is also transitive, from and , it follows that . Hence, .
Thus, we have shown that .
- Now, take any . By definition, . Since and is symmetric, , and by transitivity, from and , we get . Hence, .
Thus, .
Since we have both and , it follows that .
(2) Assume :
If , then by definition of equivalence classes, .
Conclusion:
We have shown that . Therefore, the statement is true.
Would you like more details or have any questions? Here are some related questions:
- What are some examples of equivalence relations in everyday life?
- How does an equivalence class partition a set?
- Can an element belong to more than one equivalence class?
- What are the implications of transitivity in equivalence relations?
- How can you apply equivalence relations in set theory and logic?
Tip: Equivalence classes group elements of a set that are indistinguishable under a given relation, essentially forming partitions of the set!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Set Theory
Equivalence Relations
Equivalence Classes
Formulas
-
Theorems
Properties of Equivalence Relations (Reflexivity, Symmetry, Transitivity)
Suitable Grade Level
Undergraduate Mathematics or Advanced High School
Related Recommendation
Understanding Equivalence Relations and Equivalence Classes in Set Theory
Proof that a Belongs to Its Equivalence Class in Set Theory
Proving Every Element is an Equivalence Class under an Equivalence Relation
Proof: An Element a is in its Equivalence Class [a] under Relation R
Proof: Relation Induced by a Partition is an Equivalence Relation