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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27 views

Proving Equivalence Relation $xPy$ iff $ y = x + n\pi$ on The Reals

I am trying to prove that the relation P on $\mathbb{R}$ by the rule $\forall x, y \in \mathbb{R}, xPy \text{ if and only if } \exists n \in \mathbb{Z} \text{ such that }y = x+ n\pi$ From what I can ...
0
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1answer
9 views

How many distanct equivalence classes are picked out by this relation?

Let $x\text{R}y \iff x-y=2k \quad k \in \mathbb{Z}$ How many distinct equivalence classes are there for this relation? I want to say thre are as many equivalence classes as there are integers, but ...
0
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1answer
32 views

Counting Positive Integer Divisors

Let $A$ be the set of all positive integer divisors of $3^6 5^8 11^{10} 17^{15}$. Define the relation $R$ on $A$ as follows. For $x, y \in A, xRy$ when $x | y$. Determine the number of ordered pairs ...
3
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1answer
63 views

Relationship between $S$ and $S^{-1}$

This is one of the problem I have been solving in Velleman's How to Prove book: Suppose $R$ is a relation on $A$, and let $S$ be the transitive closure of $R$. Prove that if $R$ is symmetric, ...
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5answers
47 views

How is “smaller than” defined on $\mathbb{R}$?

According to http://en.wikipedia.org/wiki/Binary_relation Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in ...
0
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1answer
24 views

What is the definition of a binary relation?

According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary ...
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1answer
15 views

Is this a transitive relation

I found this in a web site If $A = \{ 1, 2, 3\}$, then the relation $R = \{(2, 3)\}$ is not transitive. Why it is not transitive? The definition is if whenever an element $a$ is related to an ...
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3answers
44 views

Is a relation from a set $A$ to a set $B$ always a proper subset of $A\times B$?

Is a relation from a set $A$ to a set $B$ always a proper subset of $A\times B$? Or, is it possible that the relation covers the entire set $A\times B$?
2
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2answers
20 views

Proving two Recurrence Relations

I encountered this problem in the International baccalaureate Higher Level math book, and my teacher could not help. If anyone could please take the time to look at it, that would be great. I need ...
-2
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2answers
26 views

Is the following relation transititve? [closed]

Consider that the relation on $\{1,2,3,4\}$ is $\{(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)\}$ transitive. Please tell me the reason. Thanks.
0
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1answer
19 views

How would I convert this state-transition diagram into a regular expression ?

The state-transition diagram of a finite-state recogniser is as follows :
2
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1answer
61 views

Can a relation be both anti-reflexive and anti-symmetric?

Is it possible for a relation to be both anti-reflexive and anti-symmetric?
0
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1answer
18 views

How to work out the boolean matrix of a relation $S$?

I just wanted a little bit of guidance on how to work out this question in finding the boolean matrix of a relation : Consider the following Hasse diagram of a partial ordering relation $S$ on the ...
3
votes
2answers
29 views

Restricting a relation to an injective function

Given a relation between two finite sets, how can I determine whether it restricts to an injective function? The criterion or algorithm doesn't need to be constructive: It is enough to know that such ...
0
votes
1answer
13 views

Anti-symmetric or asymmetric for a relation between pairs in the set of Z x Z?

Is this anti-symmetric or asymmetric? I at first thought asymmetric because anti-symmetric would mean a = c and b = d which would not be true. But because the domain is the Cartesian product of ...
2
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3answers
113 views

Sets: How to show that relation R is not transitive?

I have a question which states: $R = \{ (t,t), (t,v), (t,z), (u,t), (u,u), (u,v), (u,x), (u,z), (v,v), (w,w), (w,z), (x,t), (x,v), (x,w), (x,x), (x,y), (x,z), (y,w), (y,y), (y,z), (z,z) \}$ ...
0
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1answer
21 views

A question on eqivalence relation

Please explain what the examiner means by asking: Also find [3,6] in Q 1 (a)(i)??
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2answers
26 views

Concrete explanation for how an equivalence relation is related to equivalence class and the notation employed

I'm fairly comfortable with the definition of what the three equivalence relations are. What I'm not comfortable and finding it above my head is how equivalence relation is closely related to ...
1
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2answers
32 views

Difference between domain and range for relations and functions?

What is the difference between the definition of domain and range for a relation, and that for a function?
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2answers
40 views

Well-defined and Equivalence relations

I am wondering why the following is well-defined... The definition of well-defined is given as; $g:(X/\sim) \to Z$ is well-defined if a mapping $f:X \to Z$ can be found where $f$ has the property $x ...
1
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1answer
17 views

Non-ordered n-tuple?

In many mathematics texts I've seen "ordered n-tuple" appear, and in such texts, there isn't any mention of just "n-tuple". So I'm wondering: are there really cases where one writes "n-tuple" and ...
0
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1answer
54 views

Proof of an equivalence relation

Let S be the relation on R defined by $xSy \Leftrightarrow x=|y|$, $\forall x,y\in\Re$ Is the relation reflexive, symmetric and/or transitive? By my proof that 1) $x=|y| \Rightarrow |y|=x$ ...
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0answers
30 views

geometric description of equivalence classes [closed]

For each of the following relations on $\mathbb{R}^{2}$, give a geometric description of the relation classes $[(0,0)]$ and $[(3,4)]$ 1) Let $S$ be the relation defined by $(x,y)S(z,w)$ iff ...
1
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1answer
47 views

Trouble Understanding a Combinatorics Problem

This question appeared on my combinatorics exam. I did not even understand the question. Determine the number of functions, $f:\{1,2,3\} \to \{1,2,3\}$, that satisfy $$f(1)+f(3)\equiv0\ (\text{mod ...
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3answers
24 views

Determine whether the relations are symmetric, antisymmetric, or reflexive.

This exercise is given in my textbook and I am trying to solve it. Determine whether they are symmetric, antisymmetric or reflexive. $R_1=\{(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)\}$ ...
1
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2answers
33 views

Equivalence relation $x^m=y^n$

Show that $R=\{(x,y) \in\mathbb{N}^2:\exists m,n \in \mathbb{N} \text{ s.t. } x^m=y^n\}$ is an equivalence relation or disprove otherwise Reflexivity and symmetricity were really easy to show ...
0
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0answers
12 views

Characterization of monovalued functions

Let $f$ be a binary relation. Let $(\bigcap G)\circ f = \bigcap_{g\in G}(g\circ f)$ for every set $G$ of binary relations. Can we prove that $f$ is monovalued (a function)?
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2answers
23 views

simple clarification of equivalence relation and order relation notation meaning

So by definition an equivalence relation is a binary relation on a subset of the cartesian product of our set A, i.e $A\times A$ satisfying the 3 conditions. But I'm a little confused about ...
2
votes
3answers
54 views

Find the inverse of $f(x,y) = (x+3y,3x+y)$

Given the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) = (x+3y,3x+y)$. Find $f^{-1}$ .( Assume $f$ is a bijection) I know how to find $f^{-1} (x) = (3x+2)$ or anything with one ...
0
votes
2answers
46 views

symmetry vs antisymmetry

So the problem I have is to write all the properties that a relation has (reflexive, symmetric, transitive, irreflexive, antisymmetric). The problem is the congruence relation on the set of triangles. ...
0
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2answers
37 views

Why is the number of subsets equal to the number of relations?

In this question I don't understand why the number of subsets is equal to the number of relations. Any help is welcome.
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2answers
36 views

Can this relation be transitive but not symmetric and reflexive?

Let $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. Give an example of a relation $T$ on $A$ with at least three elements that is not reflexive, not symmetric, but transitive. Explain clearly why ...
0
votes
2answers
43 views

Show that the function $f:\mathbb{Z} \rightarrow \mathbb{2Z}$ by $f(a)=a^3+3a+2$ is not onto

If the function $f:\mathbb{Z} \rightarrow \mathbb{2Z}$ by $f(a)=a^3+3a+2$, then show that $f$ is not onto. Hint: Show that $f(a)\neq 0$. I have a feeling I have to use the root theorem test, but I ...
3
votes
1answer
47 views

Does type-theory have a concept of “relation”?

Set theory cares about sets and relations. And then functions are relations betweens sets of inputs and outputs. Type theory, on the other hand, seems to say that there are no formal ideas of ...
0
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1answer
17 views

Prove that the relation $f$: ($\mathbb{Z^*} \times \mathbb{Z^*}$) $\rightarrow$ $\mathbb{Q}$ by $f(a,b)$ = $\frac{a+b}{a+b-3ab}$ is a function

Prove that the relation $f$: ($\mathbb{Z^*} \times \mathbb{Z^*}$) $\rightarrow$ $\mathbb{Q}$ by $f(a,b)$ = $\frac{a+b}{a+b-3ab}$ is a function I believe that f is a function and I am attempting to ...
0
votes
1answer
31 views

Is $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $f(\frac{a}{b}) = \frac{\max{(a,b)}}{\min{(a,b)}}$ a function?

Suppose that the relation $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $$ f \Bigl(\frac{a}{b} \Bigr) = \frac{\max{(a,b)}}{\min{(a,b)}} $$ is defined. Then is $f$ a function? If so, how would we prove ...
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0answers
16 views

Question Concerning a General Method to Show that a Relation is Not a Function

When showing that a relation is not a function, is there an efficient method for finding particular preimages that are mapped to more than one image, rather than attempting to find particular ...
2
votes
1answer
24 views

Intersection of two sets which are equivalence on set A is always equivalence?

If $R_{1}$ and $R_{2}$ are equivalence relations on set A ,then$ R_{1}\bigcap R_{2}$ must be equivalence relation. firstly, I am not understanding the function of R,I think that, this is only a ...
3
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2answers
30 views

why are equivalence relations called so?

"an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of ...
2
votes
0answers
24 views

Hasse Diagram Correct?

I've had to make Hasse Diagrams before, but they've always been, for lack of a better word, pretty. The lines haven't had any complicated back and forth or the like. The jump that 4 and 6 have to do ...
2
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1answer
26 views

Calculate transitive closure of a relation

I am trying to understand how to calculate the transitive closure of a set and I have read several times the definition of the transitive closure but I still cannot understand some answers I see when ...
0
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2answers
36 views

Why is this an equivalence relation, and what does the equivalence classes contain?

I'm doing some discrete mathematics exercises, but I can't seem to wrap my head around this relation: $$R(x, y) \text{ if } \exists z(\text{LiesInPart}\circ\text{LiesInCountry}(x,z) \wedge ...
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1answer
32 views

Order relations and counting the number of cases (Fubini numbers)

I have $n=2$ numbers $a$ and $b$, $a\in\Bbb{N}$ and $b\in\Bbb{N}$. Then I have the function $f$ defined as: $ f(x,y) = \begin{cases} -1, & \text{if $x<y$} \\ 0, & \text{if $x=y$} \\ +1, ...
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1answer
31 views

Binary Relations - Reflexive, Symmetric, Transitive and anti symmetric

$R$ is defined on $P(N) − \{\varnothing\}$ by $ARB$ if and only if $A \cap B \ne \varnothing$ Identify if the relation is reflexive, symmetric, transitive and anti symmetric Finding it hard to work ...
0
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1answer
17 views

Describing equivalence classes over the set of natural numbers

So for this problem we are supposed to describe the equivalence class for the following relation. $x \thicksim y$ iff $x$ mod 2 = $y$ mod 2 and $x$ mod 4 = $y$ mod 4. I am confused on what is meant ...
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1answer
19 views

verifying properties of relations to test equivalence

We are doing some more with relations and this time we are given a relation and told that it is not equivalent. We need to find out which property it does not fulfill. So we at least know there is ...
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2answers
21 views

Determining whether a relation is transitive or not.

While trying to determine whether the following relations are transitive or not, I got stuck in between. The following are the two relations - Relation R in the set $\mathbb{N}$ of natural ...
3
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3answers
135 views

Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)

Antisymmetric: $\forall x\forall y[ ((x,y)\in R\land (y, x) \in R) \to x= y]$ reflexive: $\forall x[x∈A\to (x, x)\in R]$ What really is the difference between the two? Wouldn't all antisymmetric ...
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1answer
20 views

Equivalence Relation on the set of ordered pairs of positive integers

Have a homework question, but how can I show that the given relation R is reflexive, symmetric and transitive, so that it is an equivalence relation. Appreciate assistance from anyone. "Let R be the ...
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3answers
107 views

If $R=\{(x,y): x\text{ is wife of } y\}$, then is $R$ transitive?

If $R=\{(x,y): x\text{ is wife of } y\}$, determine whether the relation $R$ is transitive or not. My Try: For Transitivity, If $(a,b) \in R$ and $(b,c)\in R\;,$ Then $(a,c)\in R.$. Here If $x$ ...