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A short class with Sarah and her students. The other ways are under More.
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Today, we'll discuss equivalence relations. Can anyone tell me what three properties an equivalence relation must satisfy?
Is it reflexivity, symmetry, and transitivity?
Exactly! We often refer to these properties as the 'RST' properties - remember that acronym? R for Reflexive, S for Symmetric, and T for Transitive. Now, can anyone give me an example of a reflexive relation?
An example could be 'a is related to a' for any element.
Great! Now, how does symmetry come into play?
If a is related to b, then b should also relate back to a.
Exactly! Now, let's move to transitivity. If a is related to b and b is related to c, what must hold?
Then a must be related to c.
Perfect! So, we have our definitions well understood. Proceeding further, let's look at unions of equivalence relations.
When we take the union of two equivalence relations, do we always get another equivalence relation? Let's explore.
Well, if both relations are reflexive, wouldn't their union also be reflexive?
Exactly! The union maintains reflexivity. Now, what about symmetry?
It should still be symmetric because if we have (a, b), we'll also have (b, a) in one of the two relations.
You're all spot on! But what about transitivity? Can anyone explain?
Just because we have (a, b) and (b, c) might not mean we have (a, c) in the union.
Right! This is why the union might fail to be an equivalence relation. A counterexample would be insightful here.
Let’s say R1 contains (a, b) and R2 contains (b, c). Without (a, c), it fails transitivity!
Exactly, excellent observation! So, unions retain reflexivity and symmetry but not necessarily transitivity.
Now, let’s switch gears and discuss intersections of equivalence relations. How do you think their properties hold up?
It sounds like they should still be equivalence relations.
Precisely! Let’s show this step-by-step. First, how do we prove reflexivity for the intersection?
If both R1 and R2 are reflexive, then (a, a) will be in both.
Exactly! Now, what about symmetry?
If (a, b) is in the intersection, it means it’s in both relations, so (b, a) is also there.
Spot on! Now for transitivity, how do we prove that?
If we have (a, b) and (b, c) in the intersection, both pairs must come from R1 and R2, so (a, c) holds.
Correct! Thus, the intersection indeed is an equivalence relation. Good work, everyone!
Let’s solidify your understanding by going through examples. Can anyone suggest a pair of equivalence relations over {1, 2, 3}?
How about R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} and R2 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}?
Great choice! What’s their union?
That would be {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}.
Is this union transitive?
No, it doesn’t include (1, 3) despite having (1, 2) and (2, 3).
Exactly! Now, let's check their intersection. What do we find?
The intersection is {(1, 1), (2, 2), (3, 3)} which is also reflexive, symmetric, and transitive.
Well done! This reinforces our findings about unions and intersections of equivalence relations.
A relation that satisfies reflexivity, symmetry, and transitivity.
Each element is related to itself.
If a is related to b, b is related to a.
If a is related to b and b is related to c, then a must be related to c.
All pairs from either relation.
Only the pairs that are in both relations.
Three properties in the mix, RST is the fix, reflexive for the self, symmetric for the rest, and transitive's the test.
Imagine a club where each member relates to themselves (reflexivity), greets others in return (symmetry), and if A likes B and B likes C, then A must like C (transitivity).
Remember RST for Equivalence: R for Reflexivity, S for Symmetry, T for Transitivity.
RST - Reflexive, Symmetric, Transitive.
Term
What are the properties of an equivalence relation?
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Define reflexivity in a relation.
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What does the union of two relations entail?
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What is the intersection of two relations?
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A relation that is reflexive, symmetric, and transitive.
A condition where every element is related to itself.
A condition where if one element is related to another, then the second is related to the first.
A condition where if one element is related to a second, and the second to a third, then the first is related to the third.
The set containing all pairs that are in either of the relations.
The set containing all pairs that are present in both relations.
This section explores equivalence relations, discussing the properties of unions and intersections of equivalence relations on a non-empty set.
In this section, we analyze two equivalence relations defined on a set and determine whether their union forms an equivalence relation, establishing that the union is reflexive and symmetric but not necessarily transitive. The section also demonstrates that the intersection of two equivalence relations is always an equivalence relation, proving reflexivity, symmetry, and transitivity.
In this section of the tutorial, we investigate equivalence relations on a set X. An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. We demonstrate this by examining two equivalence relations, R1 and R2, defined on a set X = {a, b, c}.
We first consider the union of two equivalence relations, R1 ∪ R2. While we show that this union is always reflexive (since it retains the reflexive pairs from both relations) and symmetric (as pairs in either relation will provide their symmetric pairs), it fails to be transitive in general. We illustrate this through a counterexample:
In contrast, we prove that the intersection of two equivalence relations, R1 ∩ R2, on the same set X is indeed an equivalence relation. Each of the properties is verified:
In summary, this section deeply delves into the properties of unions and intersections of equivalence relations and their implications, illustrating both successful and unsuccessful outcomes in maintaining the equivalence relation properties.