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.

learn more… | top users | synonyms

1
vote
1answer
16 views

Solution check. How many relations of equivalence $R$ are there in $\Bbb N$ that verify silmultaneously the following properties.

I'm revising some old psets while preparing an exam and going through some things that I left unverified. How many relations of equivalence $R$ are there in $\Bbb N$ that verify silmultaneously the ...
1
vote
1answer
26 views

Set of functions is not a bifunctor on Rel

Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity ...
0
votes
1answer
16 views

Relations and functionss

I am unsure how to do this, is it possible someone could give me a step by step guide so I can have a good understanding of it. f(x) and g(x) are defined over the real number set R as follows: $g(x) ...
2
votes
1answer
10 views

Count a Partial Equivalence Relations on a set

A Partial Equivalence Relation is a relation that is symmetric and transitive, and to count the number of equivalence relations on a set exists Bell Numbers, my question is How I can count the number ...
1
vote
1answer
12 views

Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence ...
1
vote
0answers
28 views

Symmetry of a relation

Let $A\neq\emptyset$ and $\mathcal{R}\subset A\times A$ a binary relation on $A$ such that the domain $D(A):=\{x\in A\mid \exists y\in A \text{ such that }(x,y)\in\mathcal{R}\}=A$ and ...
1
vote
1answer
76 views

How do you pronounce $\preceq$?

I've been reading about partial orders and partially ordered sets and have come across sentences like "Suppose that $\preceq$ is a partial order on $X$" and "If $x\preceq y$ and $y \preceq z$ then $x ...
-2
votes
0answers
19 views

how to do such type of R&F? [duplicate]

the number of functions f from {1,2,...20} onto {1,2,...20} such that f(k) is a multiple of 3 whenever k is a multiple of 4 is
0
votes
0answers
11 views

Let $A$ be a nonempty set. Prove that if $S$ and $R$ are equivalence relations on $A$, then $S \cup R$ is both reflexive and symmetric.

Let $A$ be a nonempty set. Prove that if $S$ and $R$ are equivalence relations on $A$, then $S \cup R$ is both reflexive and symmetric. My method: Let $x \in A$ be given. Then $x \in S$ or $x \in ...
3
votes
2answers
27 views

A relation $R$ is defined on $\mathbb{Z}$ by $aRb$ if and only if $2a + 2b\equiv 0\pmod 4$. Prove that $R$ is an equivalence relation.

A relation $R$ is defined on $\mathbb{Z}$ by $aRb$ if and only if $2a + 2b\equiv 0\pmod 4$. Prove that $R$ is an equivalence relation. My method: Let $a \in \mathbb{Z}$ be given. So, for any $a \in ...
1
vote
0answers
15 views

Let A be a set with $\lvert A \rvert$ = $4$. What is the max number of elements tht a relation R on A can contain so tht $R \cap R^{-1}$ = $\emptyset$

Let A be a set with $\lvert A \rvert$ = $4$. What is the maximum number of elements that a relation R on A can contain so that $R \cap R^{-1}$ = $\emptyset$? I am not sure at all how to start this ...
0
votes
1answer
14 views

Let A be the set of U. S. states. One example of a relation on A is $R = [(s,t) : s = t or s shares a border with t].

Let A be the set of U. S. states. One example of a relation on A is R = {(s,t) : s = t or s shares a border with t}. Notice that the domain of R is A, and the range of R is A. Give a different ...
0
votes
0answers
19 views

How to prove polynomial time reduction is reflexive? [closed]

How do we show that $A$ is polynomial time reducible to itself, i.e. that $A \le_p A$? I know how to prove that it is transitive, but I don't know how to prove it's reflexive.
0
votes
0answers
11 views

Determine whether the following relation is reflexive, symmetric, antisymmetric and/or transitive?

Q is defined on P(N) by aQb iff |a ∩ b| ≥ 2. I've concluded that it's symmetric, not reflexive, not antisymmetric and not transitive. Is this right?
0
votes
1answer
15 views

Relation that is reflexive, transitive, but not antisymmetric

A = {1,2} R = {(1,2)} I was just wondering if this relation meets the criteria.
1
vote
1answer
33 views

Are R,S and T equivalence relation or partial order relation?

Let $R$, $S$ and $T$ be binary relations defined as follows R is defined on $P(\mathbb{N})$ by $ARB$ if and only if $|A∩B| ≥ 2$ $S$ is defined on $Q$ by $x\mathbin{S}y$ if and only if $|x|=|y|$. ...
0
votes
1answer
18 views

What is the difference between partial order relations and equivalence relations?

From googling it i understood that a relation is both partial order relation and equivalence relation when they are reflexive, symmetric and transitive. So it appears they are the same. But as far I ...
0
votes
2answers
34 views

How to tell if the relations R, S and T are reflexive, symmetric, anti-symmetric or and transitive?

Let $R$, $S$ and $T$ be binary relations defined as follows R is defined on $P(\mathbb{N})$ by $ARB$ if and only if $|A∪B| ≥ 2$ $S$ is defined on $Q$ by $xSy$ if and only if |$x$|=|$y$|. (Note that ...
0
votes
1answer
25 views

What is the difference between total order relations and well order relations?

I know it has to be a partial order relation in order for it to be a well order relation or total order relation, but what are the differences between them.
0
votes
0answers
14 views

Determine whether this relation is reflexive, symmetric, antisymmetric and/or transitive?

R is defined on N × N (where N are natural numbers) by (a, b)R(c, d) iff a ≤ c and b ≤ d. I think it's reflexive and transitive. Not too sure about it being symmetric or antisymmetric. Any help ...
1
vote
1answer
19 views

Write out relation from a function

I have this problem: "Let $R$ be a binary relation $(x,y)\in R$ if and only if $f(x) = f(y)$ where $f: \{a, b, c, d\} \rightarrow \{0, 1\}$ given by $f(a) = 0$, $f(b) = 1$, $f(c) = 0$, $f(d) = 0$ ...
1
vote
1answer
632 views

Classify relations “is greater than or equal to”

Classify the following relations as reflexive, irreflexive, symmetric, antisymmetric or transitive. Explain each property in the context of the question. “is greater than or equal to” on the set of ...
2
votes
1answer
44 views

Functions: Relations

I'm stuck on this question and would appreciate the solution, thanks! Let $R$ and $S$ be the relations on $A = \{1,2,3,4,5\}$ given by$$R = ...
6
votes
5answers
22k views

Antisymmetric Relations

Given a set $\{1,2,3,4\}$, how is the following relation $R$ antisymmetric? $$R = \{(1, 2), (2, 3), (3, 4)\}$$ Note: Antisymmetric is the idea that if $(a,b)$ is in $R$ and $(b,a)$ is in $R$, then ...
0
votes
2answers
26 views

Prove by contradiction that If $R$ is a transitive relation on set $A$ then $R^2$ is transitive.

I saw this problem and read through it but I am still kind of confused as to what $u_1$ and $u_2$ stand for. Prove by contradiction that for a transitive relation $R$ on $A$, $R^2$ is also transitive ...
0
votes
2answers
33 views

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 ...
0
votes
0answers
30 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 ...
1
vote
0answers
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 > ...
0
votes
1answer
18 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 ...
1
vote
2answers
23 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 ...
-4
votes
1answer
33 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 ...
0
votes
0answers
29 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 ...
0
votes
1answer
18 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 ...
0
votes
1answer
12 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 ...
39
votes
4answers
5k views

Why isn't reflexivity redundant in the definition of equivalence relation?

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity. Doesn't symmetry and transitivity implies reflexivity? Consider the following argument. For any $a$ ...
0
votes
0answers
19 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
13 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 ...
0
votes
1answer
22 views

Let A, B, C be sets, with B ⊆ C. Prove that (A x B) ⊆ (A x C).

I understand why this is true but I need help answering it in a mathematical way, not just using common sense.
1
vote
2answers
17 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 ...
1
vote
1answer
18 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 ...
0
votes
2answers
25 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 ...
1
vote
2answers
29 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 ...
0
votes
2answers
32 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 ...
0
votes
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 ...
0
votes
0answers
9 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 ...
0
votes
1answer
44 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 ...
2
votes
1answer
29 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 ...