0
votes
1answer
15 views

Difference between Inclusion and continuation

Halmos defines the order continuation as follows: We shall say that a well ordered set A is a continuation of well ordered set B if B is a subset of A, if, in fact, B is an intial segment of A and ...
-1
votes
1answer
29 views

How is a relation defined on ordered sets?

I am reading that $(\mathbb{Z}, \leq )$ is a total ordered set. I understand how it satisfies reflexivity, antisymmetry, transitivity. But it says that because for any $a,b \in \mathbb{Z}$, either ...
0
votes
1answer
45 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
2
votes
1answer
48 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
0
votes
1answer
38 views

What does it mean for a binary relation to be an order on “equivalence classes” under another binary relation

I am a bit confused about this statement in "Introduction to Set Theory" by Hrbackek and Jech. The statement is as follows: "|A| <= |B| behaves like an ordering on the "equvivalence classes" under ...
1
vote
1answer
81 views

how to show a function is bijection

I have taken two numbers $p$ and $r$ where $p,r\in A = \{0,1,\ldots,4i + 1\}$ where $i\geq 1$ and $q\in B = \{0,1,\ldots,n-1\}$. Let $X$ contains all elements obtained by cartesian product of $A$ and ...
1
vote
1answer
24 views

Composition of Ordered Pair

I'm doing math exercises from a Computer Science book and I am confused as to how the following result (from the solutions manual) is obtained: Given the function f={(a,b), (a,c), (c,d), (a,a), ...
2
votes
1answer
101 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
0
votes
1answer
39 views

What is the name of this property of relation?

What is the name of property of a binary relation $R$ that states that $\lnot(a\mathrel{R} b) \iff \lnot(b \mathrel{R} a)$ for all $a, b$?
2
votes
1answer
29 views

Is this relation symmetric

$R = \{(X, Y) \in \mathscr{P}(A)^2| X \subset Y \text{ and }X \neq Y \}$ I know that $(X,Y) \in R$ holds true since $X \subset Y$. However I'm unsure if $(Y,X) \in R$ since if $Y \subset X$ then ...
1
vote
1answer
60 views

Is the subset relation on the powerset of a set, with qualification, reflexive?

I was wondering if the subset relation is reflexive? $R = \{(X, Y ) \in P(A)^2\mid X\subseteq Y \text{ and } X \neq Y \}$ I assumed they it was reflexive since for all $X \in P(A), X \subseteq X$ is ...
-1
votes
2answers
33 views

Finding Domaing and Range

Can you please tell me how i am going to solve these? $R=\{(x,y)\in \mathbb R^2 | x^2=y^2\}$ $R^{-1}=?$ $R\circ R^{-1}=?$ $\text{dom} (R)=?$ $\text{range}(R)=?$ Thanks..
5
votes
4answers
300 views

Can functions be defined by relations?

So let us say that for whatever reasons, we are not allowed to use function symbols in first-order logic. Then can we define and use a function only by relations?
2
votes
1answer
34 views

Proving that this relation is transitive

I have seen this question on a book I am reading and could not figure it out fully. The question is as follows: "Suppose A is a set, and $F\subseteq P(A)$. Let $$R_F=\{ (a,b)\in AxA|\text{ for every ...
0
votes
1answer
35 views

Proving theorems on relations

I came across the following three statements about relations. I understand why the statements are true, but I am not sure how to demonstrate them mathematically. In the following, $A,B$ are sets and ...
0
votes
2answers
35 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
2
votes
1answer
42 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
1
vote
2answers
31 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
3
votes
3answers
54 views

An example of symmetric transitive relation that is not reflexive on a set of natural numbers $\mathbb{N}$ [duplicate]

An example of symmetric transitive relation that is not reflexive on a set of natural numbers $\mathbb{N}$. My guess is that such relation does not exist, but I don't know how to prove it.
1
vote
1answer
30 views

A question on relations

Problem Statement: Let $A$ and $B$ be sets. Many books define a relation $\mathcal R$ from $A$ to $B$ to be a subset $ \mathcal R \subseteq A \times B $. Show that such an R is a ...
0
votes
3answers
119 views

What would make a function reflexive, transitive, and/or symmetric?

A binary relation $R$ is a subset of the Cartesian product between two sets $X$ and $Y$, containing a set of ordered pairs $\{(x,y) : x \in X, y \in Y\}$. $R$ is a function if each element of $X$ is ...
1
vote
3answers
43 views

How many reflexive binary relations there are on a finite countable set?

We know that binary relation is subset of Cartesian product made by set on to itself. let's say we have a set with two elements $A=\{0,1\}$ So Cartesian product is $C=A\times A = ...
0
votes
1answer
58 views

Understanding the difference between relations and functions.

$R=\{(1,2),(1,3)\}$ is a relation but not function. The logic for this is that if the first element of every ordered pair must remain different, then it is said to be function. Otherwise, it's just ...
1
vote
1answer
34 views

Finding the cardinality of $\{X\in \mathcal P(\mathbb R)| |X|=\aleph_0 \}$

Let $S$ be a relation over $\mathcal P(\mathbb R)$ such that $A,B\subseteq\mathbb R: \exists f:A\to B, \exists g: B\to A$ and $f,g$ are injections. Find the cardinality of $\{X\in \mathcal ...
1
vote
1answer
54 views

Showing $ (R ∪ R^{-1})^∗ = R^∗ ∪ R^{−1∗} $ is false by giving a counterexample.

Show that $$ (R ∪ R^{-1})^∗ = R^∗ ∪ R^{−1∗} $$ is false by giving a counterexample. I tried the following, but every time it keeps coming out as true (instead of false): If $R = \{(a,b), ...
0
votes
1answer
74 views

How can we prove these equivalence relations [closed]

We have this relation $A= (\mathbb Z_{\geq0})\times(\mathbb Z_{\geq0})\times\ldots\times(\mathbb Z_{\geq0})$. And we have the relation $R$ on $A$ such that: $(a , b)R(c , d)\iff a+d=c+b$ and ...
1
vote
1answer
24 views

Antisymmetric relation (“strong” vs “weak”)

Defining: "weak antisymmetric relation": $\forall a, b \left< a,b \right> \in R \land \left< b,a \right> \in R \Rightarrow a=b$ "Strong antisymmetric relation": $\forall a, b \left< ...
1
vote
1answer
26 views

Name for a generalized relation to be a multiset?

A relation between two sets $A$ and $B$ is a subset of $A \times B$. If taking a multiset subset of $A \times B$, e.g. allowing $(a,b)$ appears twice in the subset, is there a name for such a ...
0
votes
0answers
15 views

For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
0
votes
2answers
35 views

Greatest lower and least upper bounds for a set of pairs

Had some trouble with this question in an exam recently, and wanted to make sure I reasoned correctly. The question was: $X$ is a set of pairs of real numbers $(x,y)$, with absolute values less than ...
0
votes
2answers
42 views

Reflexive/Symmetric/Antisymmetric/Transitive

I am having issues identifying if the following are reflexive/symmetric/antisymmetric/transitive. Could anybody help me out? I have the book definitions but I'm confused on really the application of ...
1
vote
0answers
37 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
-1
votes
3answers
33 views

AntiSymmetric help

So I understand Symmetric = (a,b) > (b,a) so e.g. A = {(1,2), (2,1) = Symmetric And Anti symmetric = {(1,2), (1,1) = Anti symmetric since the (1,2) is present but no (2,1). But what if we have ...
0
votes
0answers
26 views

Relations that are closed to union and intersection

Let's define that sets are closed to union if for every relation $R,S$ that have certain traits $R\cup S$ have the same traits as well. Likewise for intersection. Determine if the following ...
0
votes
4answers
31 views

Let $(A,<_A)$ and $(B,<_B)$ be two partially ordered sets. Show that $(A,<_A)$ is not isomorphic to $(B,<_B)$.

Let $(A,<_A)$ and $(B,<_B)$ be two partially ordered sets, where $A=\{a,b,c\}$ and $<_A=\{(a,b),(a,c)\}$; $B=\{x,y,z\}$ and $<_B=\{(x,y),(y,z),(x,z)\}$. How to show $(A,<_A)$ is not ...
1
vote
1answer
38 views

state the relations of a set

Consider the following relations: ...
0
votes
1answer
31 views

Need help with compositions of relations

Prove that given relations $R_1 \subseteq A \times B$, $R_2 \subseteq B \times C$, $R_3 \subseteq C \times D$ then $(R1 \circ R2) \circ R3 = R1 \circ (R2 \circ R3)$ I don't know where exactly to ...
1
vote
1answer
36 views

List all elements of the relations

The task is to List all elements of the relations (d) S`, S-1 and S ₀T. ...
0
votes
1answer
42 views

Relations between sets

I have a mutiple choice question on finding the relation but I seem to be blanking. Can someone explain to me how this works? Let $X = \{2,3,4\}$ and $Y = \{0,1,2,3,4\}$. If a relation $P$ ...
1
vote
1answer
125 views

Exercise in Well Orderings

Prove that if $\prec$ is a well-ordering on a set X and if $Y \subseteq X$, then $\prec_Y=\{(x,y) \mid (x \in Y) \wedge (y \in Y) \wedge (x \prec y)\}$ is a well ordering on Y. I am a little ...
0
votes
1answer
41 views

Function on the set of relations: $f(R)=RK$ where $K$ is a fixed relation

$\ M$ is the set of all relations on $\ A = \{1,2,3\}$ $\ K$ is the following relation on A $\ K=\{(1,1),(2,1),(3,1)\}$ let there be $\ f :M\rightarrow M$ $\ f(R) = RK$ prove that ...
0
votes
1answer
57 views

Proving a relation is transitive

I am trying to understand transitive relations. I understand given that a set may have $\{(a,b)(b,c)\}$ it must contain $(a,c)$ for it to be transitive. But for longer sets I am getting confused in ...
1
vote
1answer
67 views

Prove that if the relation (R) is symmetric and antisymmetric on the set X, there exists a Y subset of X such that R is the = relation on Y.

Prove that for any relation R on a set X that is both symmetric and antisymmetric, there is subset Y \subseteq X for which R is the relation = on Y. I will tell you what I know. I know that ...
1
vote
2answers
39 views

How to show properties of a given relation?

I am given that R is a relation on the given set X, and I have to show if the relation is (i) reflexive, (ii) symmetric, (iii) transitive, (iv) asymmetric, and (v) give an example of an element of ...
1
vote
1answer
50 views

$M_{R^n}$; how to derive $n$ for transitive closure?

When finding the transitive closure of a relation $R$, I convert $R$ into a boolean matrix $M_R$, and find the union between $M_R$ and its powers up to $n$. $$M_{R^*} = M_{R^1} \lor M_{R^2} \lor ...
0
votes
1answer
31 views

questions about a proof to a question (about relations)

1) $R,S,T$ are relations on the same set. Prove that $R(S\cup T)=RS\cup ST$ The proof that I stumbled upon was the following: $(a,b)\in R(S\cup T)⇒((a,x)\in R)∧((x,b)\in S∨(x,b)\in T)⇒(a,b)\in ...
0
votes
2answers
23 views

Prove that R is an equivalence relation on F

A relation $R$ is defined on the set $F = \{f: \Bbb R \to \Bbb R\}$ $$fRg \iff f(0) = g(0).$$ My approach: This is reflexive because: $f(0) = f(0)$ is same as $f(0) = g(0)$ This is symmetric ...
1
vote
1answer
91 views

Determine if R is an equivalence relation

I've got this question: Consider the relationship $R$ between ordered pairs of natural numbers such that $(a, b)$ is related to $(c, d)$ (denoted by $(a, b) R (c, d)$) if and only if $ad = bc$. ...
0
votes
2answers
33 views

Use of “for all” in definition of reflexive and symmetric relations.

My book says that a relation R on A is reflexive, if $\ (a,a) \in R, \ for\ every \ a \in A$ symmetric, if $\ (a_1,a_2) \in R \implies (a_2,a_1) \in R,\ for\ all\ a_1,a_2 \in A$ Although I ...
1
vote
2answers
66 views

understanding reflexive transitive closure

Suppose I have the following relation $$R = \{(1,1), (2,3), (3,1)\}$$ To make it reflexive we add the following missing pairs: $$ \{(2,2), (3,3)\}$$ Now I wonder how to find the reflexive transitive ...