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Magazine Chapter 1: Problem 3
Given an equivalence relation \(R\) on a set \(A\), show that a. \([x] \cap[y] \neq \emptyset\) if and only if \(x \sim R y ;\) b. \([x]=[y]\) if and only if \(x \sim_{R} y\).
Short Answer
Expert verified
(a) If and only if \( x \sim_R y \). (b) If and only if \( x \sim_R y \).
Step by step solution
01
Understand the Definitions
To solve this problem, first remember that an equivalence relation \( R \) on a set \( A \) has the properties of being reflexive, symmetric, and transitive. The notation \( x \sim_R y \) means that \( x \) is related to \( y \) by \( R \). The equivalence class \( [x] \) of an element \( x \) is the set of all elements in \( A \) that are related to \( x \).
02
Prove Part (a)
We need to show that \( [x] \cap [y] eq \emptyset \) if and only if \( x \sim_R y \). **If \( [x] \cap [y] eq \emptyset \):** - There exists some element \( z \in [x] \cap [y] \). This means \( z \sim_R x \) and \( z \sim_R y \).- By the transitivity of \( R \), \( x \sim_R z \) and \( z \sim_R y \) imply \( x \sim_R y \).**If \( x \sim_R y \):**- By the definition of equivalence class, \( x \in [x] \) and \( y \in [y] \). - Since \( x \sim_R y \), every element in \( [x] \) is also in \( [y] \) and vice versa, meaning the intersection \( [x] \cap [y] \) is non-empty.
03
Prove Part (b)
We need to show \( [x] = [y] \) if and only if \( x \sim_R y \).**If \( [x] = [y] \):**- Any element \( z \in [x] \) is also in \( [y] \), meaning that \( z \sim_R x \) and \( z \sim_R y \). By choosing \( z = y \) (since \( y \in [x] = [y] \)), we have \( y \sim_R x \), so \( x \sim_R y \) by symmetry.**If \( x \sim_R y \):**- Consider any element \( z \sim_R x \). By symmetry, \( z \sim_R x \) implies \( x \sim_R z \). Since \( x \sim_R y \) and \( y \sim_R z \) by transitivity, \( z \sim_R y \). Thus, all elements in \( [x] \) are in \( [y] \) and vice versa, giving \( [x] = [y] \).
04
Conclusion
For both parts of the problem, we use the properties of equivalence relations: reflexivity, symmetry, and transitivity to prove the statements. Equivalence classes have empty intersections unless the representatives of the classes are related, which implies that the classes coincide.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equivalence Class
Equivalence classes are central to understanding equivalence relations. Given a set \( A \) and an equivalence relation \( R \), the equivalence class \([x]\) includes all elements in \( A \) that are related to a specific element \( x \) through \( R \). It's like a group of friends all connected through the same bond. If \( y \) is also in \([x]\), then \( y \sim_R x \). The class serves as a grouping tool, allowing us to categorize elements based on the relation at hand. Useful features of equivalence classes include:
- Classes are either identical or disjoint, meaning they don't overlap unless they are the same.
- Every element in the set belongs to one and only one equivalence class.
Reflexivity
Reflexivity is a key characteristic of equivalence relations. It means that every element is related to itself. Mathematically, for any element \( x \) in a set \( A \) with relation \( R \), \( x \sim_R x \). Think of this concept as self-acknowledgment鈥攅ach element strengthens its identity by being reflexively linked to itself. This property ensures that:
- No element stands alone; each is its own reference point within its equivalence class.
- Reflexivity guarantees that every class has at least one member: the element itself.
Symmetric Property
The symmetric property in equivalence relations describes a mutual relationship. If one element is related to another, then the reverse holds true as well. Formally, if \( x \sim_R y \), then \( y \sim_R x \). It's the way friendships work鈥攜ou can't be friends with someone without them being friends with you too. The symmetric property ensures that the relationship is bidirectional.
- This property creates a stable relationship between the members of an equivalence class.
- If you can get from \( x \) to \( y \), you can just as easily go back from \( y \) to \( x \).
Transitive Property
Transitivity in equivalence relations is about forming connections through intermediate elements. If an element is related to a second, which is related to a third, the first is related to the third as well. Mathematically, if \( x \sim_R y \) and \( y \sim_R z \), then \( x \sim_R z \). Think of it as a domino effect of relationships. This property allows equivalence classes to seamlessly absorb related elements:
- If \( x \) is connected to \( y \), and \( y \) to \( z \), transitivity links \( x \) to \( z \), bridging all connections.
- Transitivity helps maintain coherence and harmony within equivalence classes.
