0
votes
1answer
24 views

When proving a partial order relation is a total order do we have assume both elements are distinct?

Consider the "divides" relation on the set $A=\lbrace 1,2,2^2,.\;.\;.,2^n\rbrace$, where $n$ is a non-negative integer. Prove that this relation is a total order on $A$. First we prove $A$ is a ...
0
votes
1answer
23 views

Equivalence Relations and distinct equivalence classes

$A=\lbrace(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)\rbrace$. $R$ is defined on $A$ as follows: For all $(a, b)\;(c, d) \in A$, $(a, b) R (c, d) \iff ad=bc$ I know what they are asking but I cannot see ...
0
votes
2answers
28 views

Is there some kind of right distributivity of the subset predicate over set union?

$X \cup Y \subset Z \leftrightarrow X \subset Z \wedge Y \subset Z$. Is there a similar simple rule for $X \subset Y \cup Z$?
0
votes
0answers
26 views

some basic concepts on algebra / measure theory

I'm reading a book in Chinese on measure theory (Introduction on Measure Theory, by Yan Jia-an). In the beginning there are some algebra concepts defined that I'd like to confirm the exact meaning and ...
2
votes
5answers
126 views

Help me understand 'equivalence classes' and relations

I'm reading up on binary relations and I understand them to be a mapping from one set into another. However I'm having problems understand 'equivalence classes'. My book only gives a pretty dry ...
2
votes
4answers
74 views

Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
0
votes
2answers
58 views

An algebraic proof that a set has $2^n$ subsets. (I'm looking for an inductive argument.)

There will be duplicates of this, so let me explain why I am asking: I have become blind to what it may be, so I want hints. I am blind because I can do it "combinatorially". The question wishes me ...
3
votes
3answers
73 views

Proof: $a^2 - b^2 = (a-b)(a+b)$ holds $\forall a,b \in R$ iff R is commutative

We want to show that for some ring $R$, the equality $a^2 - b^2 = (a-b)(a+b)$ holds $\forall a,b \in R$ if and only if $R$ is commutative. Here's my proof --- I'm not sure if the first part stands ...
1
vote
2answers
31 views

How do you call this feature and property of relation?

If an operator can be defined for $k$ operands, $k \in \mathbb N$, how do you call this feature of the operator? For example, "+" is such an operator. Similarly, for a relation $R$ on a set $X$, $R$ ...
2
votes
3answers
50 views

Showing if a function is injective or surjective problem

$F : \Bbb{P}(X) \rightarrow \Bbb{P}(X) ; U \rightarrow (U-A) \cup (A-U)$ My intuition has been telling me that this function is bijective but I having the most difficult time trying to show this. Any ...
1
vote
1answer
30 views

Easy one on transitive relations

So I've got this one on my home work: Which of the following are equivalence relations on the set of $Z$ (integers)? And it presents this relation among others: $xEy$ if and only if $x^2=y^2.$ So ...
2
votes
1answer
55 views

The Empty Relation?

In elementary set theory, a relation on sets $A,B$ is usually defined as a subset of $A\times B$. We know that there are $2^{|A\times B|}$ subsets of $|A\times B|$. One of these subsets is the empty ...
4
votes
1answer
86 views

Naturally Ordered Semigroup: Why does axioms imply order of group is infinite countable ? Why are every group equal up to isomorphism?

Naturally Ordered Semigroup: http://www.proofwiki.org/wiki/Definition:Naturally_Ordered_Semigroup I know $\mathbb N$ is a naturally ordered semigroup by definition. Also I know that the naturally ...
4
votes
2answers
157 views

Is there an uncountable proper subfield of $\mathbf{R}$?

Whether there is a uncountable proper subfield of real line $\mathbf{R}$?? Thanks a lot!
1
vote
1answer
268 views

Proving that $f(A+B)=f(A)+f(B).$

Let $X$ and $Y$ denote magmas, and suppose $f : X \rightarrow Y$ is homomorphism. Then I think that for all $A,B \subseteq X$, we have $f(A+B)=f(A)+f(B).$ However, I'm not happy with my: Proof. The ...
0
votes
2answers
58 views

how to understand that products and coproducts are dual

I am reading some basic category notes, how can one relate the products to coproducts? If given a product, can one build its dual product? for example, the coordinate product $(x,y)$ : $ R \times R$, ...
2
votes
0answers
53 views

Difference between defining a constant and beginning with it in a structure

For example, let's suppose that I have my structure $\langle\mathbb{R},+\rangle$ and that $\exists!x\forall a\in \mathbb{R}(a+x=x+a=a)$ as an axiom. In this case $0:=x$. But what if I consider the ...
3
votes
3answers
121 views

Re-write $1 \cdot x$ to $x$.

Given the following bi-directional re-write rules (where $1$ is a constant, $^{-1}$ is a unary operator, $\cdot$ is a binary operator, and $x,y,z$ are arbitrary terms): $$\begin{align*} x \cdot 1 ...
3
votes
2answers
2k views

Prove that the set of all algebraic numbers is countable

A complex number $z$ is said to be algebraic if there are integers $a_0, ..., a_n$, not all zero, such that $a_0z^n+a_1z^{n-1}+...+a_{n-1}z+a_n=0$. Prove that the set of all algebraic numbers is ...
3
votes
1answer
52 views

Surjections and equivalence relations

(a) Let $f: A \to B$ be a surjective function. We define $a_1 \sim a_2$ if $f(a_1)=f(a_2)$. Prove that $\sim$ is an equivalence relation. Reflexivity: This comes for free. If $a_1 \sim a_1$, ...
2
votes
2answers
209 views

What are authoritative publications regarding foundational mathematics?

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related ...
4
votes
3answers
159 views

Groups of cardinality greater than the continuum

Can you give me some examples of groups of cardinality greater than continuum? All the (infinite) examples that I've been taught are countable or of cardinality continuum. Are there any ''natural ...
-1
votes
1answer
81 views

$\mathbb{R}/{\sim}$: A Question about the Formal Definition of a Quotient

For an equivalence relation $\sim$ what is $\mathbb{R}/{\sim}$? I mean explicitly and formally...
10
votes
2answers
246 views

Characterization properties of number sets $\mathbb{N},\mathbb{ Z},\mathbb{Q},\mathbb{R},\mathbb{C}$

When people say that a structure is defined up to isomorphism means, accordingly, that they assume certain properties that make it completely determined under certain operations and relations. ...
7
votes
4answers
238 views

How to justify the existence of a function, in general?

Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times ...
1
vote
1answer
70 views

Is there a preference between proving a total order (strict vs partial)?

I know that proving a relation $\mathcal{R}$ to be a strict total order (asymmetric, transitive,and total ) implies that the relation $S$ defined as $X\mathcal{S} Y \longleftrightarrow ...
2
votes
0answers
81 views

How to denote the set of binary relations of which a particular ordered pair is a member?

Given a universe $U$ and two subsets $S$ and $T$ (also, both members of $U$), what is the name given to denote the set of all binary relations in $U$ where the ordered pair $(S,T)$ is a member? The ...
1
vote
1answer
108 views

Dimension of $\operatorname{Hom}(V, W)$

What is the dimension of $\operatorname{Hom}(V, W)$ if at least one of the two vector spaces $V, W$ is infinite dimensional? In the sense of cardinal numbers. Thanks
1
vote
1answer
95 views

Finite ring of sets

I have some questions about finite rings of sets and I'll be very grateful for any help. Let E be some fixed non-empty set. Suppose we are given some finite ring of subsets of set E, i.e. some ...
4
votes
1answer
1k views

Counting binary operations on a set with $n$ elements

I am trying to solve following problem but not able to find any way to proceed. Let $S$ be a set having $n$ elements. Can we count about number of binary operations that can be defined on a set? Can ...
1
vote
0answers
40 views

Polynomial ring definition [duplicate]

Let $F$ be an arbitrary field (or just a ring). Usually $F[t]$ is defined as a set of all finite sequences of 'numbers' which can be multiplied etc, $F[t_1,t_2]$ is a ring of polynomials over $F[t_1]$ ...
0
votes
2answers
107 views

Show that a set is denumerable.

I have the following question: Let $A=\{x:\exists m,n \in\mathbb{Z} \text{ such that } x=m+n\sqrt{p}\}$, where $p\in\mathbb{Z}$ is a fixed prime. Show that $A$ is denumerable. I hope someone can ...
5
votes
5answers
568 views

Is there an idempotent element in a finite semigroup?

Let $(G,.)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
1
vote
3answers
135 views

Subring and field question in Z mod 10

The set R= [0],[2],[4],[6],[8] is a sub-ring of $Z_{10}$ prove that R has unity and that R is a field. I find this a bit odd but i believe [6] is the unity as opposed to the conventional [1] i am also ...
0
votes
1answer
183 views

Definition of binary operation on a set

About the definition of binary operation on a set, in my notes it says a binary operation on S is a map $*:S\times S\to S$, it does not have to be a function, it is a mapping. But in the textbook, it ...
1
vote
3answers
99 views

Proof that the $(\mathcal P (\mathbb N) \space , \space \triangle)$ is an abelian group?

I'm stuck trying to figure out how to prove that $(\mathcal P (\mathbb N) \space , \space \triangle)$ is an abelian group? I know the definition, but I'm confused how to incorporate the fact that I'm ...
0
votes
3answers
825 views

Show that a map is well-defined and homomorphism

Define a function $f_n:\mathbb{Z}_m \to \mathbb{Z}_m$ as a map $\bar{a}\mapsto n\cdot \bar{a}$. Show that it is both well-defined and a group homomorphism. For the well-defined part, I know that I ...
1
vote
2answers
120 views

Properties of set $\mathrm {orb} (x)$

Properties of set $\mathrm {orb} (x)$: ${\displaystyle \bigcup_{x\in X}\mathrm{orb}(x)=X}$; $\mathrm{orb}(x)\cap\mathrm{orb}(y)=\emptyset$ for all $x,y\in X, x\neq y$ How to prove it? Please ...
5
votes
4answers
1k views

What is the difference between a Subgroup and a subset?

What is the difference between a Subgroup and a subset? I know hardly any Abstract algebra, just some things from youtube and wikipedia, but the notion of a subgroup being part of a larger group and a ...
7
votes
1answer
230 views

Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems

There are some set systems with algebraic titles, such as "field", "algebra", "ring" and "semi-ring" (and possibly other titles), in their names. Examples are a sigma field (aka sigma algebra, ...
0
votes
1answer
125 views

Examples of dictionaries between two distinct fields of mathematics (or between “differents” structures of math).

I'd like to meet explicit examples of dictionaries between two distinct fields of Mathematics (or between two "different" structures of Mathematics). I'm not interested in the usual sense dictionary ...
5
votes
1answer
251 views

Epic implies Surjective

I'm starting to work through Algebra: Chapter 0, and I'm fumbling a bit. One of the problems in the first chapter asks me to formulate a definition for epimorphism based on having seen the definition ...
3
votes
2answers
752 views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
5
votes
2answers
1k views

Given the Hasse diagram tell if the structure is a lattice

Let's consider the following Hasse diagram: I need to tell whether this is a lattice. By lattice definition I can prove the above shown structure $M_5$ to be a lattice if and only if $\forall x,y ...
2
votes
2answers
142 views

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$?

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? Is it that of the continuum? Proof?
2
votes
2answers
653 views

Complete set of equivalence class representative

Let $\sim$ be a relation on $\mathbb{R}$ and $x\sim y$ if and only if $x-y\in \mathbb{Z}$. (a) Show that ~ is an equivalence relation (b) Give a complete set of equivalence class representatives. ...
2
votes
1answer
124 views

Countability of $\mathbb{Q}$

I have seen a few proofs which shows the countability of rationals (denoted $\mathbb{Q}$). But they always involve picking a representation for the rationals (like fractions,Calkin-Wilf trees) and ...
4
votes
2answers
173 views

Which algebraic structure captures the ordinal arithmetic?

Consider the set class $\mathrm{Ord}$ of all (finite and infinite) ordinal numbers, equipped with ordinal arithmetic operations: addition, multiplication, and exponentiation. It is closed under these ...
2
votes
5answers
594 views

General questions about equivalence classes and partitions

1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with ...
1
vote
2answers
283 views

Draw Hasse diagram as two elements have same image

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: $$\begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) ...