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Interactive Audio Lesson
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Union of Equivalence Relations
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Today we'll explore the union of two equivalence relations. Can anyone tell me what an equivalence relation is?
An equivalence relation is a relation that is reflexive, symmetric, and transitive.
Correct! Now, here's a key point: If you have two equivalence relations, their union is always reflexive and symmetric. But is it always transitive?
No, right? In some cases, it might not be transitive.
Exactly! For instance, let's consider the relations R1 and R2 over the set X = {a, b, c}...
We can observe through a counterexample that R1 ∪ R2 can fail to have the transitivity property.
So the union can create issues with transitivity but not with reflexivity or symmetry?
Indeed! Remembering the acronym 'RST' for Reflexive, Symmetric, but not necessarily Transitive can help you.
To sum up, while the union of two equivalence relations is always reflexive and symmetric, it may not be transitive.
Intersection of Equivalence Relations
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Now, let's shift gears and discuss the intersection of two equivalence relations. How does this behavior differ from their union?
I think the intersection is also an equivalence relation, isn't it?
You're spot on! Let's prove this. Both R1 and R2 are reflexive, so what about R1 ∩ R2?
R1 ∩ R2 will definitely also be reflexive because each element will relate to itself in both R1 and R2.
Right! What about symmetry?
If (a, b) is in the intersection, then it must also be in both R1 and R2. And since those are symmetric, (b, a) must also be in both of them.
Correct! And how about transitivity?
If (a, b) and (b, c) are in the intersection, they must also be in both R1 and R2. So, (a, c) must also be there.
Exactly! Thus, R1 ∩ R2 is an equivalence relation. Remember the phrase 'PIE' - for Properties Include Equivalence.
Composition of Equivalence Relations
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Let's delve into the composition of two equivalence relations now. What do we need to show if we're asked whether R1 ∪ R2 equals R1 ◦ R2?
We need to show that the composition must also hold the same properties as the union, right?
Yes, and we demonstrate the sufficiency of transitivity. Let's provide a general proof by breaking it down into cases.
What are the various cases we consider for (a, b) and (b, c) in the union?
Great question! We need to check all configurations. If both pairs come from R1, they hold. If they come from R2 or they are mixed, we can show continuity.
So we are utilizing transitivity from the individual relations to validate for the union!
Exactly! And understanding this leads to deeper insights into connectivity in mathematical relations!
In summary, the conditions under which the composition and the union equate leads to notable implications regarding equivalence relations.
Introduction & Overview
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Quick Overview
Standard
This section elaborates on the characteristics of two equivalence relations over a non-empty set, discussing the conditions under which the union is not necessarily an equivalence relation while the intersection is. It further delves into implications regarding compositions.
Detailed
Detailed Summary
In this section, we delve into the properties of equivalence relations, specifically examining operations involving two equivalence relations over a non-empty set. The union of two equivalence relations is analyzed, showcasing a counterexample where the union fails to satisfy transitivity, even though it remains reflexive and symmetric. The interpretation here is that while both individual equivalence relations exhibit reflexivity and symmetry, their union does not guarantee transitivity.
In contrast, the intersection of the same two equivalence relations is proven to always remain an equivalence relation since it continues to satisfy all three fundamental properties: reflexivity, symmetry, and transitivity. The implications of the union and intersection of equivalence relations lead us into a conjoint understanding of the conditions under which one may maintain equivalence in relation to compositions, guided by the equality of the composite and the union of equivalence relations.
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Understanding Equivalence Relations
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In this question, you are given two equivalence relations on a non-empty set X. You are asked to prove or disprove whether R1 ∪ R2 is an equivalence relation or not. It turns out that a union of two equivalence relations need not be an equivalence relation. To demonstrate this, consider a counter-example where X = {a, b, c} and the equivalence relations R1 and R2 are defined as:
R1 = {(a, a), (b, b), (c, c), (a, b), (b, a)}
R2 = {(a, a), (b, b), (c, c), (b, c), (c, b)}.
It is easy to see that both of these relations are equivalence relations over the set X as they satisfy reflexivity, symmetry, and transitivity.
Detailed Explanation
This chunk focuses on the importance of equivalence relations and their unions. Equivalence relations are defined by three properties: reflexivity (every element is related to itself), symmetry (if a is related to b, then b is related to a), and transitivity (if a is related to b, and b is related to c, then a is related to c). Here, we illustrate that while each individual relation, R1 and R2, satisfies these properties, their union R1 ∪ R2 might not satisfy all properties. Specifically, even though both R1 and R2 are reflexive and symmetric, the union is not necessarily transitive. An example illustrates this point clearly: the presence of (a, b) and (b, c) in the union without (a, c) results in a violation of transitivity.
Examples & Analogies
Consider a social network where R1 represents friendships and R2 represents coworkers. Just because you are friends with someone and someone else is a coworker does not mean you are directly connected to the second person. In other words, knowing someone through different relationships does not guarantee a direct tie between you and all of their connections.
Characteristics of the Union of Equivalence Relations
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The union R1 ∪ R2 is reflexive over the set X because ordered pairs (a, a), (b, b), and (c, c) are present in this relation. It is also symmetric since if (a, b) is in the union, then (b, a) is also present.
However, it is easy to see that the union is not a transitive relation. For instance, if (a, b) is in R1 ∪ R2 and (b, c) is in R2, (a, c) is not necessarily in the union. Hence, generally, while the union of two equivalence relations is always reflexive and symmetric, it need not be transitive.
Detailed Explanation
This chunk assesses the characteristics of the union of two equivalence relations. Even when we take the union of R1 and R2, we still find that this union maintains reflexivity and symmetry. However, it fails the transitivity test, which is crucial for establishing if this union can be classified as an equivalence relation. Thus, we confirm the mixed properties of this union: guaranteed reflexivity and symmetry, but questionable transitivity.
Examples & Analogies
Think of this situation in the context of class relationships: being in the same study group (R1) and participating in a club together (R2). Just because you study with Sarah and Sarah is in the club with John doesn’t mean you and John are connected directly in any way. While you both share a connection through Sarah (reflexive), and you both know Sarah (symmetric), without a direct link you are not guaranteed any simple relationship (transitive).
Definitions & Key Concepts
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Key Concepts
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Union of equivalence relations: Demonstrates that the union is not guaranteed to be transitive.
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Intersection of equivalence relations: Always results in an equivalence relation.
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Composition of equivalence relations: Links the understanding of the connection between unions and composite relations.
Examples & Real-Life Applications
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Examples
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Example of equivalence relations R1 = {(a,a), (b,b), (c,c), (a,b), (b,a)} and R2 = {(a,a), (b,b), (c,c), (b,c), (c,b)} showing their union is not transitive.
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Example illustrating that the intersection of R1 and R2 yields another equivalence relation, confirming reflexivity, symmetry, and transitivity.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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Reflexive, Symmetric too, Transitive will get a view. In any union we may lose, Transitivity, that's the dues.
📖 Fascinating Stories
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Once in a math land, two relations met. Their union was grand, but failed the transitive pet. They shook hands with an intersection, forming a trio's perfection!
🧠 Other Memory Gems
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Remember 'RST': Reflexive, Symmetric, but beware when Transitive does not see!
🎯 Super Acronyms
Use 'PIE' to recall
- Properties Include Equivalence for intersections that stand tall.
Flash Cards
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Glossary of Terms
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Term: Equivalence Relation
Definition:
A relation that is reflexive, symmetric, and transitive.
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Term: Reflexive
Definition:
A relation R on a set S is reflexive if every element is related to itself.
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Term: Symmetric
Definition:
A relation R is symmetric if for all a, b in S, if aRb, then bRa.
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Term: Transitive
Definition:
A relation R is transitive if for all a, b, c in S, if aRb and bRc, then aRc.
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Term: Union
Definition:
The union of two sets contains all elements that are in either set.
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Term: Intersection
Definition:
The intersection of two sets contains all elements that are common to both sets.
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Term: Composition
Definition:
The composition of two relations is defined when a relation of the first and a relation of the second act together.