# Tagged Questions

For reflexive, symmetric and transitive relations. Use it with the tag (relations).

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### Finding equivalence classes in a set of functions with a condition

I can't seem to work around this problem... Let $n$ and $m$ be two positive integers. $F$ is the set of functions from {1,..,n} to {1,...,m}. We define the relation $R$ as: $f R g$ if and only if ...
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### Does $\lim\limits_{x\rightarrow a}(f-g)(x)=0$ then $f(x)\sim_{x\rightarrow a}g(x)$ and revert?

Why does $\lim\limits_{x\rightarrow a}(f-g)(x)=0$ then $f(x)\sim_{x\rightarrow a}g(x)$ false and $f(x)\sim_{x\rightarrow a}g(x)$ implies $\lim\limits_{x\rightarrow a}(f-g)(x)=0$ false? I was given ...
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### Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
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### Explain this step in solving this system of linear congruences.

I'm looking at this example and it doesn't make sense to me. We have to solve the following systems of linear congruences : $x\equiv 1\pmod 5$ $x\equiv 2\pmod 6$ $x\equiv 3\pmod 7$ We take ...
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### Equivalence relation on $\mathbb{N}$ with infinitely many equivalence classes

Does there exist an equivalence relation, defined on $\mathbb{N}$, with infinitely many equivalence classes, all of which contain infinitely many elements? I see no reason for such a relation not to ...
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### Relations and restriction of a function.

This is a homework question: "Let R be an equivalence relation on a set S. For A ⊆ S, we define RA to be the restriction of R to elements of the set A, i.e., RA is a relation on A such that for any a,...
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### Equivalence relations on intersections

Let $R_1$ and $R_2$ be equivalence relations on a set $S$. Their intersection is on the relations considered as sets of ordered pairs. Identify the equivalence classes of the relation $R_1 \cap R_2$. ...
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### Which language L have exactly one equivalence class

Consider the alphabet {a,b}, for which language does the equivalence relation R have exactly one equivalence class? From what i understand about equivalence class, each state is consider a class. So ...
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### How to describe an equivalence class?

For example: the relation given is $x\sim y$ if $f(x)=f(y)$. What do you have to say when describing a equivalence class?
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### equivalence relation and sets

The question: Let $X$ and $Y$ be two sets, and let $S$ be an equivalence relation on set $X$ and $T$ be an equivalence relation on set $Y$. Define a relation $R$ on $X ×Y$ by $(a,b)R(c,d)$ if and ...
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### Equivalence classes in the logical equivalence on some finite set of propositional formulas

I'm having trouble understanding the following problem: Let $S_n$ be the set of all formulas that can be built up with the atoms $\{A_1,...,A_n\}$. How many equivalence classes does the ...
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### Equivalence relations and composition, intersections of them

I've been having some trouble with this one, I hope someone get's it. Let S and R be equivalence relations within X. Prove that if R∘S is an equivalence relation, then it is equal to the ...
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### Lagrange's theorem intuition

I cannot grasp the intuition behind |G|/|H|=[G:H]. Starting from the equivalence relation x~y if and only if x^(-1)*y is in H, I can see a sort of division, but in my mind, the equivalence relation ...
Let $A$ be a set. $P_{1}$,$P_{2}$ Are partitions of $A$. Let $E^P$ be the equivalence relation associated to a partition $P$ $$E^P:=\{\langle a,b\rangle \mid \exists \mathcal B\in P\ (b\in\mathcal B\... 2answers 39 views ### Prove that the union of two equivalence relations on the same set an equivalence relation iff? Let R and E be equivalence relations on set A. Prove that R\cup E is an equivalence relation on set A iff for all a\in A, [a]_{R} \subseteq [a]_{E} OR [a]_{E} \subseteq [a]_{R}. ... 1answer 30 views ### What is an equivalence class of an equivalence relation? I might be interpreting this wrong but in my book it says: If ~ defines an equivalence relation on A then the set of equivalence classes of ~ form a partition of A. To me, this means that the set ... 1answer 26 views ### Proved that if R\cup E equivalence relation so a / E \subseteq a / R OR a / R \subseteq a / E Let R, S be equivalence relations on A. Proved that if R\cup E Is an equivalence relation on A So \Rightarrow For all a\in A ,$$a / E \subseteq a / R$$OR$$a / R \subseteq a / E$$I ... 1answer 42 views ### Can R=\{(1,6),(2,7),(3,8)\} be said transitive? Given a relation R=\{(1,6),(2,7),(3,8)\}. It is clear that it is not reflexive and symmetric but can we say that it is transitive? 1answer 21 views ### Determinate the quotient topology I was trying to find the quotient topolgy for the next example: Let R be the real numbers with the usual topology (\tau) and define the relationship \mathcal{R} over R as follows, a \mathcal{R}... 0answers 13 views ### Determine the given relation is Equivalence Relation or not. R_{1} \oplus R_{2} I know that R_{1} \oplus R_{2} = R_{1} \cup R_{2} - R_{1} \cap R_{2}, and R_{1} \cup R_{2} is not necessarily an equivalence relation but R_{1} \cap R_{2} is always an ... 2answers 16 views ### How do I prove symmetry of a relation given a function? Let G be a group. For all g\in G , define the function f: G → G that sends x to gxg^{-1}. Define the relation ~ on G by a~b if a = f(b) for some g\in G. Prove that ~ is an equivalence relation.... 6answers 74 views ### Why yC_1x \iff yC_2x implies C_1 = C_2? C_i is a relation. Here is the text from the book Topology by Munkres: Studying equivalence relations on a set A and studying partitions of A are really the same thing. Given any partition \scr D of A, there ... 1answer 15 views ### Counting ordered pairs in quotient set We have equivalence relation E on the set$$A = \{1,2,3,4,5\}$$So the quotient set:$$A/E = \{\{1,2,3\}, \{4\}, \{5\}\}.$$How much orderd pairs we can find in E? How to count the ordered pairs?... 0answers 46 views ### LEN-Model equivalency Starting position is a principal-agent-model with incomplete information (moral hazard) and the following properties: Agent utility: u(z)=-e^{(-r_az)} Principal utility: B(z)=-e^{(-r_pz)} Effort ... 2answers 31 views ### What is the equivalence class of the following equivalence relation? If we have a equivalence relation R=_{def} \{((x_1,y_1),(x_2,y_2)) ~~| ~~x_1-y_1=x_2-y_2 \} \subseteq \mathbb{R}^2 \times \mathbb{R}^2  What is the equivalence class [(0,1)]_R? I thought it ... 1answer 123 views ### Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y) f:X→Y x,y ∈ X,xRy iff f(x) = f(y) Show that R is an equivalence relation on X. Also when X = Y = \mathbb{R} and f: \mathbb{R} \to \mathbb{R}  with x \mapsto x^2 for all x∈R find the ... 2answers 111 views ### Prove that the relation on X given by x\sim y if f(x)=f(y) is an equivalence relation Let f:X\to Y be a function between two sets. Prove that the relation on X given by x\sim y if f(x)=f(y) is an equivalence relation. I know that it should have 3 cases, reflexive (for all x... 1answer 27 views ### Stuck on equivalence relations Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) \in R if and only if ad = bc. What are the equivalence classes of this relation? I am completely ... 0answers 21 views ### equivalence class of kernel relation of floor function By taking particular values of k, I found that equivalence class of k, [k]=\{k,k+1,k+2,...,2k-1\}, equivalence class of 2k, [2k]=\{2k,2k+1,2k+2,...,3k-1\} and so on, but how to present it ... 0answers 10 views ### Number of symmetric relations Let set A=\{1,2,3\} Find number of symmetric relations that can be defined on A containing ordered pairs (1,2) and (2,1) is? Can someone give me some hint for this question? 1answer 40 views ### Show that either [x] = [y] or [x] and [y] are disjoint? An equivalence class [x]_R is defined by [x]_R = [y ∈ X : xRy]. In this proof I'm supposed to start by assuming [x] and [y] are not disjoint. Therefore, there is some element z that is in [... 4answers 155 views ### How do you show one way equivalences in mathematics? In the real world, it seems you can often take a set of things and create new things with them. Let's say you want to make bread, for example. Normally, you can create bread by putting together flour, ... 5answers 27 views ### Show that x is an element of its own equivalence class? If R is an equivalence relation on X and x is an element of X, the equivalence class is defined as [x]_R = [y ∈ X : xRy]. Since x is equivalent to itself, doesn't that automatically make ... 1answer 38 views ### Relations and their fallacies I need to find a flaw in the proof of the following statement: Any relation that is symmetric and transitive is also reflexive. False Proof: aRb implies bRa,by symmetry. Then by transitivity, aRa ... 2answers 54 views ### Proving an equivalence relation on a \mathbb Z\times \mathbb Z I am having some trouble understanding how to prove that a given binary relation is an equivalence relation. I understand that to prove that a relation is an equivalence relation, you must prove ... 1answer 38 views ### Which of the following equivalence classes are equal? Let R be the relation of congruence modulo 3. Which of the following equivalence classes are equal. [7], [-4], [-6], [17], [4], [27], [19] The answer is: [7]=[4]... 0answers 15 views ### Is Congruence Class an example of Equivalence Class? (Understanding the definition through Examples) Would you say a congruence class such as [0]_{4}, [1]_{4}, [2]_{4}, and [3]_{4} an example of Equivalence classes since: the set of all elements of \{..., -8, -4, 0, 4, 8,...\} is related to [... 2answers 38 views ### What does it mean for f([x])=[2x] for a function mapping R/Z to R/Z? Let X=R/Z (the circle), with a map f : X → X given by f([x]) = [2x]. I'm a little lost on what f([x]) = [2x] means. I thought the function was mapping the equivalence class [x] to the ... 0answers 17 views ### Find equivalence classes A is set of all propositional expressions on the propositions p1 , p2 , . . . , pn. Relation R on A is defined as (P, Q) ∈ R if P ↔ Q is a tautology. I think it is an equivalence relation, what should ... 1answer 30 views ### Equivalence Classes of an Equivalence Relation Confusion (definition and solution included) The Definition of Equivalence Classes of an Equivalence Relation is given as: Suppose A is a set and R is an equivalence relation on A. For each element a in A, the equivalence class of a, ... 1answer 26 views ### Showing an equivalence relation Let [n] denote the set of all permutations. Let R be the relation$$i_1 i_2\cdots i_n=j_1 j_2\cdots j_n if and only if there exists a $k$ such that $j_1 j_2 \cdots j_n=i_k i_{k+1}\cdots i_n i_1 ... 1answer 123 views ### Equivalence classes under logical equivalence by 13 valuations Let L be the set of 5 propositional variables. Under the equivalence relation given by logical equivalence, how many equivalence classes of propositional terms are given the value TRUE by 13 ... 2answers 114 views ### Why not just define equivalence relations on objects and morphisms for equivalent categories? My question is regarding this blog post https://unapologetic.wordpress.com/2007/05/30/equivalence-of-categories/ which I will paraphrase below: The author gives the example of a category$\mathscr{C}$... 1answer 48 views ### definition of the directed colimit of a functor Let$(\Lambda,\le )$be a directed set, which we can understand as a small category: The set of all objects is$\Lambda$and for$\lambda,\lambda '\in \Lambda$there exists an unique morphism$i_{\...
Let $X$ be the set of all nonempty subsets of $\{1, 2, 3\}$. Then $X= \{\{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$ Define a relation $R$ on $X$ as follows: For all $A$ and \$...