What Does Equivalence Relations Mean in Set Theory?

[kingstar]

Hi,

I'm reading a book on sets and it mentions a set B = {1,2,3,4}
and it says that
R3 = {(x, y) : x ∈ B ∧y ∈ B}
What does that mean? Does that mean every possible combination in the set?

Also the book doesn't clarify this completely but for example using the set B say i had another set

R = {(1,2),(2,3),(1,3),(1,1),(2,2),(3,3),(4,4)},

Would this be clarified as transitive and reflexive? My question is does a set need to have all transitive properties and all the reflexive properties to be called transitive and reflexive.

If i had another set:

R1 = {(1,2),(2,3),(1,3),(1,1),(2,2),(3,3)}

In which i removed (4,4) would this set R1 still be considered reflexive?

Thanks in advance

 

[disregardthat]

Your first example is a transitive and reflexive relation. A relation is transitive and reflexive if it satisfies the axioms for transitivity and reflexivity.

Your other example is not reflexive, since 4 is an element of X, but 4 ~ 4 is not satisfied.

 

[HallsofIvy]

My question is does a set need to have all transitive properties and all the reflexive properties to be called transitive and reflexive.

You appear to have the wrong idea about the "transitive" and "reflexive" properties. You cannot talk about "all the transitive properties" and "all the reflexive properties" because there is only one of each. We apply the term "reflexive" to the whole relation, not individual pairs. If we have a relation on set A, then it is a subset of AxA, the set of all ordered pairs with each member from set A. Such a relation is called "reflexive" if and only if, for every a in A, (a, a) is in the relation. If a particular such pair, say, (x, x), is in the relation, we do NOT call that pair "a reflexive property". Similarly, a relation is called "transitive" if and only if whenever pairs (a, b) and (b, c) are in the relation, so is (a, c). We do NOT apply the term "transitive" to the individual pairs.

 

[kingstar]

Ahh, thanks! This helped me understand it a lot better

 

[CellCoree]

Hi there! I can help clarify the concept of equivalence relations for you. An equivalence relation is a type of relation between elements in a set that satisfies three properties: reflexivity, symmetry, and transitivity. In simpler terms, this means that for any element x in a set, it must be related to itself (reflexivity), if x is related to y then y must also be related to x (symmetry), and if x is related to y and y is related to z, then x must also be related to z (transitivity).

In your first example, R3 is a set of ordered pairs (x,y) where both x and y are elements of the set B. This means that every element in B is related to every other element in B, making it a reflexive, symmetric, and transitive relation.

In your second example, R is also a reflexive, symmetric, and transitive relation as it satisfies all three properties. However, a set does not need to have all three properties to be called reflexive or transitive. In your third example, R1 is still considered reflexive because it satisfies the reflexivity property, even though it does not have all three properties.

I hope this helps clarify the concept of equivalence relations for you. Let me know if you have any other questions!