# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X \subset Y \cup Z$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X$, $R$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Is there an uncountable proper subfield of $\mathbf{R}$?

Whether there is a uncountable proper subfield of real line $\mathbf{R}$?? Thanks a lot!
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### 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 ...
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### 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 $(x,y)$ : $R \times R$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $a_1 \sim a_1$, ...
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### 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 ...
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### 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 ...
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### $\mathbb{R}/{\sim}$: A Question about the Formal Definition of a Quotient

For an equivalence relation $\sim$ what is $\mathbb{R}/{\sim}$? I mean explicitly and formally...
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### 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. ...
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### 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 ...
### 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