1
vote
3answers
73 views

Set containing all rings!

Does there exist a set containing all rings ? Possible idea :I think such set is not possible.If S is a set containing all rings i think we can again define a structure on S to make it Ring and that ...
2
votes
1answer
38 views

Is algebra over a set also algebra over a field?

During my studies I have come across two different notions of the term "algebra", namely algebra over a set and algebra over a field (the field its vector space always being Euclidean space in my ...
0
votes
3answers
34 views

Powerset bijection problem

Please do not provide a full answer for this. Let $2^{S} = \{f : S \rightarrow \{0, 1\}\}$. For $A \subseteq S$, define $\chi_{A}\in2^{S}$ by $$\chi_{A}(s) = \begin{cases} 0 & \text{if } s ...
0
votes
2answers
55 views

Doubt pertaining to this Equivalence Relation.

$1$. True or false? If $R$ is an equivalence relation on a set $S$ and it has only finitely many equivalence classes altogether, then $S$ itself is a finite set. I think the answer is true ...
0
votes
1answer
120 views

The group of bijections of an infinite set that move only finitely many elements

If someone can prove the following and give me an easy to understand explanation I would appreciate it! I have tried what omnomnomnom has suggsetd in the hints, but can't figure this out. A(S) is ...
1
vote
2answers
88 views

What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)

In general an operation is a function $f:S^n \to S$, where $n$ denotes the arity of the operation, and a nullary operation is a function $f:S^0 \to S$. It is clear that $S^n=S \times S \times \cdots ...
0
votes
1answer
27 views

Every non-empty subset of the integers which is bounded above has a largest element.

I was reading a proof about every non-empty subset of the integers which is bounded above has a largest element, but i have troubles in one step. Here is the proof: Since $S$ is a non-empty subset of ...
1
vote
1answer
53 views

Clarify Cartesian Products and Binary Operations

So tell me if I'm saying this write. A Cartesian Product is a function f:X x Y --> Z , where some unknown structural operation on the sets X and Y produces a set Z as its codomain, and Z is a set of ...
0
votes
3answers
35 views

How many algebras of subsets of $X$ contain exactly four elements?

Let X be a set with five elements. How many algebras of subsets of X contain exactly four subsets? Well $\emptyset, X$ must be in any algebra of subsets of $X$ so that means we have to find two more ...
2
votes
2answers
77 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
0
votes
2answers
48 views

Sets as extremely trivial groups

A group is a structure defined upon an underlying set which is endowed with a single binary operator that has some rules attached to it. I was wondering whether one could describe a set itself as ...
0
votes
2answers
24 views

Relation between the euclidean space and a set of functions.

Let $n$ be an integer. In what sense can $\mathbb{R}^n$ be seen as the collection of functions $\lbrace n\to \mathbb{R}\rbrace$? (-what is $n$ here?) And also, does this (bijection of sets, I guess?) ...
0
votes
1answer
38 views

Major sets and their algebra, operations

There are many major sets of numbers which have definitions we are all familiar with. For example, the set $\mathbb{Z}$ contains a countable number of elements which are given the labels ...
3
votes
1answer
93 views

Infinite Direct Sums Vs. Infinite Direct Products

Let $|R|=|S|=\infty$. In very many concrete categories, I know $R^S$ can be identified as the set of all functions from S to R, and the much "smaller" $R^{\oplus S}$ can be identified as the subset of ...
1
vote
2answers
65 views

Why is this not a $\sigma$ - algebra

Why is $\displaystyle \{\cup_{i=1}^{\infty} (a_i, b_i] : a_i < b_i, i=1,...,n, n \in \mathbb{N_0} \}$ not a $\sigma$-algebra? It looks ok to me, the only way I can see that this would fail is ...
0
votes
2answers
54 views

Does an injective function imply two sets have same cardinality?

In the book "A first course in abstract algebra" by John B. Fraleigh, he states in the introduction chapter (which deals with sets and relations) that for two sets $X$ and $Y$, they have the same ...
0
votes
0answers
51 views

Would like to confirm answer (regarding sets)

As you might know from my precious questions, I am pretty weak with quantifiers. Below is my solution to the stated problem, if incorrect, could someone explain why? My attempted solution: ...
4
votes
4answers
68 views

Number of join-irreducible elements of a lattice: is it monotonic?

Let $\mathcal L$ be a sub-lattice of $\mathcal P(X)$, where $X$ is a finite set. Denote by $\mathcal I(\mathcal L)$ the set of union-irriducible elements of $\mathcal L$ (i.e. $A\in \mathcal ...
0
votes
1answer
121 views

Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
0
votes
0answers
26 views

Operation table of Hasse diagram

Consider the following Hasse diagram: My book gives the following join and meet operation tables for this diagram: $$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ ...
1
vote
1answer
27 views

Question on Cardinality ..Help

a) Let $n$ be a positive integer. Define a relation on $\mathbb{Z} $, which yields a partition of $\mathbb{Z}$ with $n$ elements; and give the partition. b) Deduce that $n\omega = \omega$ where ...
0
votes
1answer
41 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
32 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
31 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
46 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
161 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
78 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
1answer
80 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
127 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
57 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
31 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
127 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
107 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
174 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
406 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
66 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
56 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
122 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 ...
8
votes
2answers
3k 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
60 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
216 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
179 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
82 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
250 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
292 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
74 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
86 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
111 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
104 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 ...