This tag is intended for questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric,...), composition of relations and similar stuff. More-or-less the things about relations taught in the first elementary set theory or discrete math course.

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Equivalence relation and equivalence classes given function and relation

Given a function $f : A → B$, let $R$ be the relation defined on $A$ by $aRa′$ whenever $f(a) = f(a′)$. Prove that $R$ is an equivalence relation and determine the equivalence classes. To prove that $...
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31 views

Which of the following are equivalence classes?

For example 37, I've determined which ones are equivalence relations but am having trouble on example 37: 1-7 determining which of the following are equivalence classes. I'm having trouble ...
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13 views

Symmetric and reflexive closure on positive integers

Finding the symmetric and reflexive closure of the relation $R = \{(a, b) | a > b\}$ on the set of positive integers. For symmetric closure I have: $R = \{(a, b) | a > b\} U \{(b, a) | a > b\...
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2answers
29 views

Counting number of relations that are symmetric and reflexive.

I've looked at the other two problems similair to mine but I'm having a bit of an issue understanding as their solutions seems a bit more complex. While I for the most part understand my professors ...
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0answers
33 views

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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1answer
35 views

Let f be the function from $A = \{a, b, c, d\}$ to $B = \{e, f, g, h\}$ given by $f = \{(a,e), (b,f), (c,g), (d,g)\}$. [closed]

Let $f$ be the function from $A = \{a, b, c, d\}$ to $B = \{e, f, g, h\}$ given by $f = \{(a,e), (b,f), (c,g), (d,g)\}$. If $D = \{a,b\}$, what is $f(D)$? If $G = \{f,g\}$, what is $f^{-1}(G)$? If $...
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1answer
14 views

Confirming my understanding in determining if a relation is reflexive, symmetric, or transitive

I think I have a grasp on how to determine if a relation is reflexive, symmetric, or transitive. Just to make sure I understand it correctly, if I have the following relation: for $(a,b) \in \mathbb{...
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1answer
20 views

Can you use constants from the domain in a First Order Formula? [closed]

Say I have a First Order Signature defined like so: $N = (\{1,2,3\dots\},T)$ Where T is a binary relation symbol. Can I use values from the domain to define functions over this signature? For ...
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0answers
20 views

Maximize function over a a set with a transitive and antisymmetric relation

Let $\mathcal{R}$ be a transitive and antisymmetric relation defined over a finite set $X$. For any set $S\subseteq X$ let $\Gamma(S)=\left\{y\in S ,\not \exists x\in S \mid (x,y)\in\mathcal{R}\...
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1answer
22 views

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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0answers
11 views

Determining if Poset based on Domain and Comparison Operator?

Can someone help me with how to think about the below problem? I know that a poset is a relation which is reflexive, antisymmetric, and transitive, but unless I'm dealing with finite sets I have a lot ...
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1answer
15 views

Determine if given relation is an equivalence relation, and describe the partition?

Can someone walk me through how to do these types of problems in my Discreet Mathematics II Textbook? (ex.): Determine whether the given relation is an equivalence relation on the set. Describe the ...
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1answer
3 views

Finding pairs with respect to lexicographic order that meet a condition from a set?

I am working some problems out of my textbook for Discrete Mathematics II and was wondering if someone could tell me how to think through and go about solving the following type of problems (there are ...
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2answers
19 views

Question about the exclusive or operator

Let $R_1$ be the “less than” relation on the set of real numbers and let $R_2$ be the “greater than” relation on the set of real numbers, that is, $R_1 = \{(x, y) | x < y\}$ and $R_2 = \{(x, y) | x ...
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2answers
28 views

How to find inverse of a relation if the inverse isn't a function?

I am trying to find the inverse of the following function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}$ given by $f(a)=\frac{(-1)^a(2a-1)+1}{4}$. I switched $x$ and $y$ and then tried solving for $y$. This ...
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2answers
45 views

define a relation $R$ on $S$?

Let S be the set of humans. 1) Define a relation $R$ on $S$ that is reflexive, symmetric, and transitive but not antisymmetric 2) Define a relation $R$ on $S$ that is symmetric and antisymmetric ...
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0answers
11 views

Clarification on a reflexive function

$R_1 = \{(a, b) | a ≤ b\}$ is a reflexive function, but I'm confused on why it is. $a≤b$ but doesn't that not necessarily mean that there is an $a$ that will equal $b$? Couldn't all of the $a$ very ...
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2answers
31 views

Confusion on sets and relations

I'm confused on how the number of subsets equals the number of relations. If set A = {1, 2} then AxA would be {}, {1}, {2}, {1,2}. I'm confused on how there are $2^{n^2}$ subsets of $A$ x $A$ because ...
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0answers
15 views

Understanding Fuzzy Composition operations

There are two common forms of composition operation in Fuzzy Theory: max–min composition max–product composition Let R be a relation that relates elements from ...
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1answer
30 views

For a given relation in $\mathbb{N}\times \mathbb{N}$ find the number of elements in it's equivalence class

The whole problem goes like this: We define the relation $R$ in $\mathbb{N}\times \mathbb{N}$ in the following way: $(a,b)R(c,d)$ iff $a-d=c-b$ First find proof the it's a relation of equivalence (...
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1answer
74 views

A relation that is both reflexive and irrefelexive

I didn't know that a relation could be both reflexive and irreflexive. However, now I do, I cannot think of an example. So what is an example of a relation on a set that is both reflexive and ...
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1answer
41 views

Why is $x \mid y$ over $\Bbb N$ a partial order but not total order?

I understand why $x \mid y$ is an example of a partial order relation over $\Bbb N$. But can someone explain why its not a total order relation? By definition a total order relation on a set $A$ is a ...
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1answer
31 views

Why is the Relation R3 Transitive?

Given $A = \{1,2,3,4\}$ in the Relation $\mathcal{R} = \{(1,1),(2,2),(3,3),(4,4)\}$ I understand why $\mathcal{R}$ is Reflexive, Symmetric but why is it also transitive? In my understanding for a ...
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1answer
25 views

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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21 views

Binary relations between any two sets

I have some doubts regarding relations and binary relations in particular.This is what I understand : 1) The graph $G_R$ of a relation R on X and Y is the subset of X × Y defined by $G_R$ = {(x, y) ∈...
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1answer
27 views

Deteremining whether the relation R on the set of all Real Numbers is Reflexive, Symmetric, Antisymmetric, Transitive, and/or Irreflexive

I am attempting to work out a problem from my Discreet Mathematics Textbook and am a little stuck on part of this one question. I was wondering if someone could walk me through (b) and (c) on the ...
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1answer
20 views

How many $(a,b)\in\Bbb Z^+\times\Bbb Z^+$ are there so that $(a,b)\mathrel{S}(n,n)$?

Let $n \in \mathbb Z^+ = \{1,2,3,\dots\}$. How many $(a,b) \in \mathbb Z^+ \times \mathbb Z^+$ are such that, $(a,b)S(n,n)$ where $$(a,b)S(c,d) \iff a + b ≤ c + d$$ In our case, $(a,b)S(n,n) \iff ...
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1answer
19 views

Is following Relation from $X$ to $Y$ is a Relation or not?

$$X=\{1,2,3,4,5\} \text{ and } Y=\{1,3,5,7,9\}$$ $$R=\{(x,y)\mid y=2+x,x\in X, y\in Y\}$$ my textbook says it is a relation from $X$ to $Y$. But for $x=2$, $y=4$ but $4$ is not in $Y$. How is that ...
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2answers
47 views

Definability of the $<$ order relation on the natural numbers using addition. [closed]

Show that the usual order relation $<$ on the natural numbers is definable in the structure $(\mathbb{N}, +)$ with only addition. My teacher has clarified this for me and quantifiers can be used. ...
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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 ...
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1answer
20 views

Give an example that the following condition does not imply WARP

I know how to prove that Weak Axiom of Revealed Preference (WARP) implies the following condition: if $a\in B_1, B_1 \subseteq B_2, a\in C(B_2)$, then $a\in C(B_1)$. $C$ here is a notation for choice ...
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2answers
31 views

A property of reflexive transitive closure

Suppose $R$ is a binary relation on a set $S$. Let $R^+$ be the reflexive transitive closure of $R$. That is, $R^+$ is minimal relation which includes $R$ and is both transitive and reflexive. By ...
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1answer
141 views

Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite. I understand that ...
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1answer
21 views

Rewrite “x > a” in Iverson brackets as Heaviside function

Let's say I have a Heaviside function defined like this: $$ H(x) = \begin{cases} 0, \text{ if } x < 0\\ 1, \text{ if } x \geq 0 \end{cases} $$ Then I have a so called Iverson brackets: $$ [...
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45 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 ...
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1answer
27 views

Prove that if $(a, b) \in \rho$ then $[a]_{\rho} = [b]_{\rho}$

Given an equivalence relation $\rho$ over the set $A$. Prove that if $(a, b) \in \rho$ then $[a]_{\rho} = [b]_{\rho}$. Also prove that if $(a, b) \notin \rho$ then $[a]_{\rho} \cap [b]_{\rho} = \...
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0answers
20 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 ...
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1answer
70 views

Suppose that R and S are reflexive relations on a set A. Prove or disprove each of these statements.

I am doing this question with my own attempt. Can anyone help me with the formal way of proving? Thanks! Suppose that $R$ and $S$ are reflexive relations on a set $A$. Prove or disprove each of ...
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1answer
13 views

Determine whether each of these combinations of R 1 and R 2 must be an equivalence relation.

I have this question but not really sure how to do it when there is union and interception symbol. I am easily confuse when this 2 symbol appear. From my understanding I know that equivalence relation ...
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35 views

Let $R_1$ and $R_2$ be the “congruent modulo 3” and the “congruent modulo 4” relations, respectively. Find $R_1\cap R_2$ and $R_1 \cup R_2$.

I have this question and would need help on how to find $R_1 \cup R_2$. My working for $R_1 \cap R_2$ is shown below: Let $R_1$ and $R_2$ be the “congruent modulo 3” and the “congruent modulo 4” ...
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1answer
37 views

Determine whether the relation is reflexive, symmetric, anti-symmetric, and/or transitive?

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if a) everyone who has visited Web page a has also ...
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1answer
28 views

What are the values of the sum?

what are the values of $\sum_{j \in S} 1$, where S = {1, 3, 5, 7}. if we have $\sum_{j = 1}^{n} 1$ then the answer will be n. But what happens if this a set?
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0answers
9 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?
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33 views

How Galois connections between powersets correspond to binary relations?

How to show the well-known bijective correspondence between Galois connections (or rather polarities) between two powersets on some (fixed) sets with binary relations between these sets? You can also ...
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1answer
24 views

Can a tuple be defined as an instance of a relationship?

SO let's say x and y are in relation. Is (x,y) a tuple and an instance of a relation ? Is a n-tuple an instance of a n-relation?
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1answer
15 views

What is a minimal relation?

I was reading this definition of transitive closure of a relation, where is written that the transitive closure is minimal: the transitive closure of a binary relation R on a set X is the ...
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1answer
50 views

A book or Source to further study Relations

I have completed a course on Discrete Mathematics and really enjoyed studying the chapter on relations. In fact I went back and finished what we hadn't covered in class. I did basic stuff like n-ary ...
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2answers
27 views

Having problem with finding the number of ordered pairs.

Y and Z are proper subset of X this means that X is having all the elements of Y and Z and also Y$\ne$X and Z$\ne$X (This is because we are talking about proper subset and not just subset). Let Y={1} ...
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1answer
22 views

Listing the elements of an equivalence class

Say you have x = {1,2,3,4,5}, y ={2,5}, and c = {2,3} and the relation R: ARB iff AUY = BUY ...
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21 views

Hasse Diagrams and partially ordered sets

3) Define U = {1, 2, 3, 4, 5}. Consider the following subsets of U: P = {2, 3, 5}, O = {1, 3, 5}, E = {2, 4}, S = {3} a) Create the Hasse diagram using $\subseteq $ as the partial order on sets E, ...