What is Equivalence relations: Definition and 60 Discussions
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation.
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class.
So, my approach and solution are as follows:
$$
[x * y] = \{ z \in M : z \sim (x * y) \}
$$
Since we know that ##a * b \sim a^{\prime} * b^{\prime}## we can rewrite ##z## as ## x^{\prime} * y^{\prime} ##. Plugging this in yields:
$$
[x * y] = \{ x^{\prime}, y^{\prime} \in M : x^{\prime} *...
Dear All,
I am trying to solve the attached two questions.
In both I need to determine if the relation is an equivalence relation, to prove it if so, and to find the equivalence classes.
In both cases it is an equivalence relation, and I managed to prove both relations are reflexive. Now I...
Sometimes to help describe one expression, another expression is shown that produces identical results. The exact equivalence of expressions is indicated with ‘ ≡’.
For example: rot90 ([1, 2; 3, 4], -1) ≡ rot90 ([1, 2; 3, 4], 3) ≡ rot90 ([1, 2; 3, 4], 7)
What is the meaning of 'rot90;?
What...
(a) I present the following counter example for this. Let ##A = \{0,1,2,\ldots \}## and ##B = \{ 2,4,6, \ldots \} ##. Also, let ##C = \{ 1, 2 \} ## and ##D = \{3 \}##. Now, we can form a bijection ##f: A \longrightarrow B## between ##A## and ##B## as follows. If ##f(x) = 2x + 2##, we can see...
I understand that the first part of the equation is an equivalence class due to reflexivity, symmetry, and transivity... but I am confused on the second part. Could someone please help me out? THANKS
<Moderator's note: Moved from a technical forum and thus no template.>
Not sure this should be under Linear and Abstract Algebra, but regardless I need help with a question in my mathematical proofs course.
Here it is:
Let ∼ be a relation defined on Z by x ∼ y if and only if 5 | (2x + 3y).
(a)...
Hello all,
I have a few questions related to the different number of equivalence classes under some constraint. I don't know how to approach them, if you could guide me to it, maybe if I do a few I can do the others. Thank you.
Given the set A={1,2,3,4,5},
1) How many different equivalence...
Homework Statement
For each of the relations defined on ℚ, either prove that it is an equivalence relation or show which properties it fails.
x ~ y whenever xy ∈ Z
Homework EquationsThe Attempt at a Solution
Here's my problem: I am starting off the proof with the first condition of...
Homework Statement
For the set ℤ, define ~ as a ~ b whenever a-b is divisible by 12. You may assume that ~ is an equivalence relation and may also assume that addition and multiplication of equivalence classes is well defined where e define [a]+[ b ] = [a+b] and [a]*[ b ] = [ab] for all [a],[ b...
Consider this self-evident proposition: "If object A has the same mass as object B and object C separately, then object B has the same mass as object C." Why isn't this stated as a law, but the zeroth law of thermodynamics is?
Is there a physical quantity u such that the u of A is equal to the...
Hello,
I've to construct a proof of the following statement: Prove that if S is a set and S_1... S_k is a partition of S, then there is a unique equivalence relation on S that has the S_i as its equivalence classes.
I'm really not sure how to go about this proof at all, so any help would be...
Prove that the following is an equivalence relation on the indicated set. Then describe the partition associated with the equivalence relation.
1. In Z, let m~n iff m-n is a multiple of 10.2. The attempt at a solution
Reflexive: m-n = 0
0 ∈ Z, and 0 is a multiple of every number...
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...
Hi guys! First time poster, long time lurker! I can't make any sense out of equivalence relations:confused: These kinda questions crop up every year on the exam and I was wondering if someone could help me understand the concept behind them.
(i)Show that relation R defined on the of the
set S =...
Homework Statement
Let ## H = \{ 2^{m} : m \in Z\}##
A relation R defined in ##Q^{+} ## by ##aRb ##, if ## \frac{a}{b} \in H##
a.) Show that R is an equivalence Relation
b.) Describe the elements in the equivalence class [3].
The Attempt at a Solution
For part a, I think I am able to solve...
Hey everyone, I have three problems that I'm working on that are review questions for my Math Final.
Homework Statement
First Question: Determine if R is an equivalence relation: R = {(x,y) \in Z x Z | x - y =5}
and find the equivalence classes.
Is Z | R a partition?
Homework...
If {{a,b},{c}} is the partition of {a,b,c}. When finding the equivalence relation used to generate a partition, is it enough to say {a,b}x{a,b} U {c}x{c}?
Thanks
Andy
Homework Statement
The question is let E1 and E2 be equivalence relations on set X. A new relation R is defined as the E1 o E2, the composition of the two relations. We must prove or disprove that R is an equivalence relation.Homework Equations
The Attempt at a Solution
I know that we must...
Show that if R1 and R2 are equivalence relations on a set X, then R1 is a subset of R2 iff every R2-class is the union of R1 classes.
Attempt: I don't understand that if R2 has elements nothing to do with the elements of R1, how can an R2 class be a union of those elements belonging to an R1...
Our math Teacher asked us to find how many equivalence relations are there in a set of 4 elements, the set given is A={a,b,c,d} I found the solution to this problem there are 15 different ways to find an equivalence relation, but solving the problem, i looked in Internet that the number of...
Our math Teacher asked us to find how many equivalence relations are there in a set of 4 elements, the set given is A={a,b,c,d} I found the solution to this problem there are 15 different ways to find an equivalence relation, but solving the problem, i looked in Internet that the number of...
I am studying set theory and I came across various definitions like equivalence relations, partial order relations, antisymmetric and many more. I am aware mathematicians don't care about real life applications but still - why are we defining so many relations? What is the use of defining...
Let G be a group and let H be a subgroup of G.
Define ~ as a~b iff ab-1εH.
Define ~~ as a~~b iff a-1bεH.
The book I am using wanted us to prove that each was an equivalence relation, which was easy. Then, it asked if these equivalence relations were the same, if so, prove it. My initial...
Homework Statement
Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack
a) { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) }
This one is not reflexive
Homework Equations
I understand that...
Homework Statement
prove that if a~a' then a+b ~ a' + b
Homework Equations
The Attempt at a Solution
I can prove that if a=a' then a+b = a' + b but how can I apply this to any equivalence relation
I'm not sure if I did these 2 questions correctly, so would someone please check my work for any missing ideas or errors?
Question 1:
Homework Statement
Prove:
For every x belongs to X, TR∩S(x) = TR(x) ∩ TS(x)
Homework Equations
The Attempt at a Solution
TR(x) = {x belongs to X such that...
Homework Statement
Deciede if the following are equivalence relations on Z. If so desribe the eqivalence classes
i) a\equiv b if \left|a\right| = \left|b\right|
ii) a\equiv b if b=a-2
Homework Equations
The Attempt at a Solution
i) \left|a\right| = \left|a\right| so its...
Let Q be the group of rational numbers with respect to addition. We deﬁne a
relation R on Q via aRb if and only if a − b is an even integer. Prove that this is an
equivalence relation.
I am very stumped with this and would welcome any help
Thank you
1. Let R be a relation on X that satisfies
a) for all a in X, (a,a) is in R
b) for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R.
Show that R is an equivalence relation.
2. In order for R to be an equivalence relation, the following must be true:
1) for all a in X, (a,a) is...
Homework Statement
For each of the relations on the set R x R - (0,0) (ie. no origin) :
- prove it is an equivalence
- give the # of equivalence cases
- give a geometric interpretation of the equivalence cases assuming an element of R x R is a point on a plane
a) {((a,b),(c,d)) |...
Homework Statement
Provide an example that shows why the reflexive property is not redundant in determining whether a relation is an equivalence relation or not. For example, why can't you just say, "If xRy then yRx by symmetric property, and then using transitive property you get xRx."...
Homework Statement
Let X = {a,b,c,d}. How many different equivalence relations are there on X? What subset of
XxX corresponds to the relation whose equivalence classes are {a,c},{b,d}
Homework Equations
N/A
The Attempt at a Solution
So I wrote out all the possible "blocks"...
I have two questions:
i) Does a distinct equivalence relation on a set produce only one possible partition of that set?
ii) Can multiple (distinct) equivalence relations on a set produce the same partition of that set? In other words, given a set S and two distinct equivalence relations ~...
Homework Statement
Given is the set X. The set of functions from X to [0,1] we call Fun(X,[0,1]). On this set we consider the relation R. An ordered pair (f,g) belongs to R when f^{-1}(0)\setminus g^{-1}(0) is a countable set.
a) Prove that R is transitive.
b) Is R an equivalence relation...
We have a equivalence relation on [0,1] × [0,1] by letting (x_0, y_0) ~ (x_1, y_1) if and only if x_0 = x_1 > 0... then how do we show that X\ ~is not a Hausdorff space ?
x,y,z\in\mathbb{R}
x\sim y iff. x-y\in\mathbb{Q}
Prove this is an equivalence relation.
Reflexive:
a\sim a
a-a=0; however, does 0\in\mathbb{Q}? I was under the impression
0\notin\mathbb{Q}
Symmetric:
a\sim b, then b\sim a
Since a,b\sim\mathbb{Q}, then a and b can expressed as...
1) Recall that an equivalence relation S on set R ( R being the reals) is a subset of R x R such that
(a) For every x belonging to R (x,x) belongs to S
(b) If (x,y) belongs to S, then (y,x) belongs to S
(c) If (x,y) belongs to S and (y,z) belongs to S then (x,z) belongs to S
What is the...
Homework Statement
I have got myself very confused about equivalence relations. I have to determine whether certain relations R are equivalence relations (and if they are describe the partition into equivalence classes, but I'll worry about that once I understand the first part).
Here are...
Homework Statement
I need a little help in understand this question:
Let E and F be two sets, R a binary relation on the set E and S a binary relation on the set F. We define a binary relation, denoted RxS, on the set ExF in the following way ("coordinate- wise"):
(a,b) (RxS) (c,d) <-->...
Hey!
Hoping you guys could help me with a small issue. No matter how hard I try, I don't seem to fully understand the notion of an equivalence relation, and henceforth an equivalence class. What I do understand that, in order to have and equivalence relation, it is defined to satisfy three...
I wasn't sure whether to post this in the algebra forum or here, but it seems that this is more of a logic question so I'm going with here. I am trying to understand whether there is a difference between the following two definitions of an equivalence relation:
Definition 1: A binary relation...
This is a question from A consise introduction to pure mathematics (Martin Liebeck)
Hi guys, just stuck on one problem was wondering if someone could lend me hand.
Let ~ be an equivalence relation on all intergers with the property that for all "m" is an element of the set of intergers ...
This is a question from A consise introduction to pure mathematics (Martin Liebeck)
Hi guys, just stuck on one problem was wondering if someone could lend me hand.
Let ~ be an equivalence relation on all intergers with the property that for all "m" is an element of the set of intergers ...
Statement:
Prove or Disprove: A relation ~ on a nonempty set A which is symmetric and transitive must also be reflexive.
Ideas:
If our relation ~ is transitive, then we know: a~b, and b~a \Rightarrow a~a.
Therefore our relation ~ is reflexive, since b~c and c~b \Rightarrow b~b, and c~a...
Hello!
I'm a bit lost on these questions pertaining to equivalence relations/classes. If someone could run me through either, or both, of these questions, I'd be very thankful! I'm completely lost as to what to do with the z in terms of set S...
Homework Statement
Show that the given...
A relation R on R^2 is defined by (x_{1},y_{1})\mathit{R}(x_{2},y_{2})\;\;\;if\;\;\;x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2}
How do you show that R is an equivalance relation?
Homework Statement
Hi I have justed switched to a new subject and have some question.
1) Show that if X is a topology space then there exist an equivalence relation if and only if there exist a connected subset which contains both x and y.
2) Show that the connected components are a...
Homework Statement
On set PxP, define (m,n)\approx(p,q) if m*q=p*n
Show that \approx is an equivalence relation on PxP and list three elements in equivalence class for (1,2)
Homework Equations
The Attempt at a Solution
I will appreciate any help how to start this problem...
Homework Statement
Find relations that satisfying
just Reflexive
just Symmrtic
just Transitive
(R) & (S), but not (T)
(R) & (T), but not (S)
(S) & (T), but not (R)
Homework Equations
S=Z
(a,b) \inR if <=> a>b (T) but, not (S) & (R).
the ex is given in the class, but...