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A short class with Sarah and her students. The other ways are under More.
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Today, we will learn about equivalence relations. Can anyone tell me what makes a relation an equivalence relation?
It has to be reflexive, symmetric, and transitive, right?
Exactly! To remember these properties, think of the acronym RST: Reflexive, Symmetric, Transitive. Can you give me examples of each property?
For reflexive, every number is equal to itself?
Very good! Now, for symmetry, if a is related to b, then b must be related to a as well. And for transitivity, if a is related to b and b to c, then a must relate to c. Can anyone summarize these properties?
All must be true for it to be an equivalence relation!
Exactly! Great summary.
Let’s look at a concrete example of an equivalence relation. Consider the relation defined on integers where a is equivalent to b if they give the same remainder when divided by a fixed modulus m. Who remembers how we express this?
By saying a ≡ b (mod m).
Right! If we take modulo 3, what would the equivalence classes look like?
[0] would include all multiples of 3!
And [1] would include integers like 1, 4, 7, right?
Exactly! [1] includes integers of the form 3k + 1. And what about [2]?
That would be 3k + 2, like 2, 5, 8, and so on.
Great job! Remember, these classes have unique properties: they are either the same or completely disjoint.
Now, let’s explore the properties of equivalence classes. What do you know about them?
Every equivalence class is non-empty.
That's right! Each class always contains at least the number itself. Can you give examples of what [0] would look like under mod 3?
[0] = {…, -6, -3, 0, 3, 6, …}!
Excellent! And how do we illustrate that equivalence classes cannot overlap?
If an element belongs to both classes, then they must be the same class.
Exactly! If [a] intersects [b], then [a] must equal [b].
A relationship satisfying reflexivity, symmetry, and transitivity.
Group of elements related to a specific element by the equivalence relation.
Each element relates to itself.
If a relates to b, then b relates to a.
Linking three elements - If a relates to b and b relates to c, then a relates to c.
Reflexive, symmetric, transitive too, that's how equivalences can be true!
Imagine a classroom where students can only relate to those sitting next to them, each maintaining their groups. No student can belong to two groups at once, creating clear equivalence classes.
Remember RST for relations: R for Reflexive, S for Symmetric, and T for Transitive.
Use the acronym REM for Remember Equivalence Modulus to help recall how classes are formed in mod arithmetic.
Term
Equivalence Relation
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Term
Equivalence Class [x]
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Term
Mod m
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A relation that satisfies reflexivity, symmetry, and transitivity.
A subset of a set containing all elements related to a specific element under an equivalence relation.
A property of a relation where every element is related to itself.
A property of a relation where if one element is related to another, the reverse is also true.
A property of a relation where if one element is related to a second, and the second is related to a third, then the first is related to the third.
This section explores equivalence classes defined by equivalence relations, demonstrating their properties and providing practical examples using modulo arithmetic.
In this section, we define equivalence relations and their associated classes, demonstrating how integers relate under modulo operations. The section showcases examples of equivalence classes and their properties, including reflexivity, symmetry, and transitivity.
This section delves into equivalence classes, which arise from equivalence relations defined on sets. An equivalence relation on a set A satisfies three properties: reflexivity, symmetry, and transitivity. Without these properties, a relation cannot be classified as an equivalence relation.
An example is provided where integers (ℤ) are examined under the relation of congruence modulo m, specifically noting that two integers a and b are equivalent if they yield the same remainder when divided by m. The equivalence class of an element a, denoted [a], is a set of all elements in A that relate to a through the equivalence relation.
In demonstrating with modulo 3, we find equivalence classes such as [0] containing all multiples of 3, [1] containing all integers of the form 3k + 1, and [2] for integers of the form 3k + 2. It is concluded that equivalence classes for any integer modulo relations either coincide or are completely separate.