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A short class with Sarah and her students. The other ways are under More.
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Welcome everyone! Today, we will explore whether the union of two equivalence relations results in another equivalence relation. Can anyone tell me what properties define an equivalence relation?
I think they are reflexivity, symmetry, and transitivity.
Exactly! Now, we can confirm that the union of two equivalence relations is always reflexive and symmetric. But what about transitivity? Let's consider our example with the sets.
That's interesting! But could you clarify why it might not be transitive?
Sure! If we have two relations, say R1 and R2, and we find that (a, b) is in their union and (b, c) is present, for the union to be transitive, (a, c) must also be there. However, there are cases where it isn't.
So we need to find a counterexample to display that?
Exactly! Remember our sets X = {a, b, c} and the relations R1 and R2? Let’s summarize that diffraction in properties of the union!
Now that we've discussed the union, let’s move on to the intersection of two equivalence relations. Who can tell me if the intersection also forms an equivalence relation?
I think it would! Each relation is reflexive, right?
Correct! We can prove that the intersection indeed remains reflexive, symmetric, and transitive. Let's break it down together. First, because both relations are reflexive, (a, a) will be present in the intersection.
What about symmetry?
Good question! Since (a, b) is in the intersection, symmetry applies due to how both initial relations were defined. Lastly, can anyone summarize how transitivity holds?
If (a, b) and (b, c) are in both relations, then (a, c) must also exist?
Exactly! Now you clearly see how intersections maintain the equivalence relation properties!
Let’s discuss when the union of two equivalence relations equals their composition. This is crucial for understanding their behavior together.
Is that an if and only if statement?
Correct! We need to establish both directions of implication. How do we start proving the first part?
We assume the composition equals the union, right?
Exactly! Show that the transitive property holds under these premises. What about the reverse implication?
That must mean if the union is an equivalence relation, then their composition must equal the union!
Correct! Keep practicing this concept until it becomes second nature.
To finish, let's look at the function P(n). This function counts the number of equivalence relations over a set of n elements. Anyone have an idea how to relate it with partitions?
I think P(n) represents the same number of partitions, right?
Exactly! Each equivalence relation corresponds to a unique partition. How can we express P(n) in terms of previous values?
Maybe using a recurrence relation with combinatorial coefficients?
Spot on! By examining j elements that can pair with the first element, we express the recurrence relation involving C(n-1, j). Great job today everyone. Let’s summarize our learning!
Defined by reflexivity, symmetry, and transitivity.
The combination of two sets, including all their elements.
The common elements of two sets.
A method of combining relations where transitivity must hold.
Counts the number of equivalence relations over a set of n elements.
Consider the set X = {a, b, c} with 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)}. The union R1 U R2 is reflexive and symmetric but not transitive.
The intersection R1 ∩ R2 retains reflexivity since it includes (a,a), (b,b), (c,c), and maintains symmetry and transitivity.
Reflex, Symm, Transitive too, Equivalence relations are true blue!
Imagine a club where each member knows themselves (reflexive), knows each other (symmetric), and if member A knows B, and B knows C, then A surely knows C (transitive).
Remember 'RST' for Reflexive, Symmetric, Transitive when you think of equivalence relations.
Use 'E-RST' to remember Equivalence requires Reflexivity, Symmetry, and Transitivity.
Term
What is reflexivity?
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Property needed for a relation to be symmetric
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Define the composition of two relations.
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A relation that is reflexive, symmetric, and transitive.
An element is related to itself.
If one element is related to another, then the second is related to the first.
If one element is related to a second, and that second is related to a third, the first is related to the third.
The set of elements common to two sets.
The set that contains all elements from both sets.
Combining two relations such that if one element is related to another, and that second element is related to a third, the first element relates to the third.
Denotes the number of equivalence relations over a set of n elements.
This section explores equivalence relations, specifically examining their unions and intersections, and discusses conditions under which these operations yield equivalence relations.
The section delves into the properties of equivalence relations, focusing on how the union and intersection of two equivalence relations may or may not maintain these properties. It presents definitions, proofs, and counterexamples to clarify when these operations result in equivalence relations, emphasizing the importance of reflexivity, symmetry, and transitivity in determining the nature of the relations.
In this section of the chapter, we investigate the interplay between two significant equivalence relations over a non-empty set. We begin by addressing whether the union of two equivalence relations maintains the property of being an equivalence relation. Through a counterexample involving specific sets, we demonstrate that while the union is always reflexive and symmetric, it does not necessarily maintain the transitive property.
In contrast, we prove that the intersection of two equivalence relations is always an equivalence relation, affirming the preservation of reflexivity, symmetry, and transitivity. Furthermore, we explore a condition under which the union of two equivalence relations can be equivalent if and only if their composition equals their union, providing proofs for both implications.
We conclude by defining the function P(n) that denotes the number of equivalence relations over a set of n elements, and exploring its recurrence relations through combinatorial arguments, further solidifying the connection between equivalence relations and partitions.