Suppose that R1 and R2 are equivalence relations on the set S . Determine whether each of these

Suppose that R1 and R2 are equivalence relations on the set...

Key Concepts

Symmetric Difference of Equivalence Relations

The symmetric difference of two relations includes those pairs that belong to one relation or the other, but not to both. This operation can disrupt the transitivity property, although reflexivity and symmetry might be preserved under certain conditions. As a result, the symmetric difference of two equivalence relations does not necessarily form an equivalence relation.

Intersection of Equivalence Relations

The intersection of two equivalence relations consists of only the pairs that are common to both relations. Since both equivalence relations already satisfy reflexivity, symmetry, and transitivity, their intersection inherits these properties, ensuring that the intersection is always an equivalence relation.

Union of Equivalence Relations

The union of two equivalence relations takes all pairs that are related in either relation. However, although each relation is reflexive and symmetric, their union might not be transitive. Thus, the union of two equivalence relations does not necessarily yield an equivalence relation.

Equivalence Relations

An equivalence relation is a binary relation on a set that satisfies three properties: reflexivity (every element is related to itself), symmetry (if one element is related to another, then the second is related to the first), and transitivity (if one element is related to a second and the second to a third, then the first is related to the third). These properties lead to a partition of the set into equivalence classes.

Reflexivity, Symmetry, and Transitivity

These are the fundamental properties that any equivalence relation must have. Reflexivity ensures every element stands in relation to itself, symmetry guarantees that the relation is bidirectional between any two related elements, and transitivity allows the relation to extend over chains of related elements, making the structure robust enough to form well-defined equivalence classes.

Set Operations on Relations

Operations such as union, intersection, and symmetric difference combine or modify relations and can affect whether the resulting relation still possesses the properties required for an equivalence relation. Understanding how these operations impact reflexivity, symmetry, and transitivity is crucial in determining if the resulting relation will satisfy the criteria for being an equivalence relation.

Transcript

00:01 This video we are asked to determine whether the following combination of these two equivalent relations are also equivalent relation.

00:15 So we only know that these two are some equivalent relation on the set s, on the same set.

00:23 And this statement is very general, so it's for any set as, any set as, any, equivalent relation are 1 and r2 so there are two choice or two scenario that can happen either the combination we are looking at is equivalent relation in which case we have to prove the three conditions of equivalent relation right and another scenario is that is not an equivalent relation in that in this case we only need one counter example to to justify saying that is not true all right so in those cases i will give counter example which mean we can i can decide on what kind of set s is and what relation are one and are two are okay so let's go so you you see the can't return on the board you you probably know that it's not yes this one is not equivalent relation why so i have this scenario where suppose s has three elements abc r1 and r2 are equivalent relations you have everything related to itself so we have reflexive symmetry because if ab is in there, b is in there.

02:20 And transitivity, but we don't have to really check that because we create it in such a way that it doesn't concern three things at once.

02:31 So there is no ac or something like that.

02:36 But the main thing to consider is that when we union them, now inside, inside, inside this union we have ab and b c, right? and so transitivity would imply that we must have ac as well for this thing to be, this union to be equivalent relation.

03:06 But no, we don't have this element because it's not in any like equivalent relation that we're starting with.

03:19 So this one is not.

03:24 This is one counter example...

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