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14

In formal set theory, the closest thing to a definition of a "set" we get is "something which every objects either belongs to or doesn't belong to" -- in other words, if you have any object, you can ask the set whether the thing you have in your hand is one of its member or not and it will answer either yes or no. And it will give you the same answer each ...

7

No. The phrase "vacuously true" is used informally for statements of the form $\forall a \in X: P(a)$ that happen to be true because $X$ is empty, or even for statements of the form $\forall a \in X: Q(a) \to P(a)$ that happen to be true because no $a \in X$ satisfies $Q(a)$. In both cases, it is irrelevant what statement $P(a)$ is. I guess you could turn ...

7

They're two different uses of the term 'contains'. While the English the meanings in each is the same, the first example you give refers to the membership relation $x\in S$, which is to say that $x$ is an element of $S$, while the second example you give refers to the subset relation $A\subset B$, which is to say that every element of $A$ is an element of ...

5

The principle of permanence is best observed as the (often unmentioned) guiding light when extending the realm of "number". We start with $\mathbb N$ and addition and multiplication and observe that certain rules hold. Among these are associativity and commutativity of addition and multiplication, the distributive law, cancellation (i.e. $a+b=a+c$ implies ...

4

These functions naturally arise in everyday problems. For example polynomials are just a consequence of adding and multiplying. The exponential function has the peculiar property that it is its own derivative, and in some way this property ("eigenfunction under the derivative ioperator") is the typical cause for it appearing. The logarithm is just the ...

4

This is a book from the 19th century. You should not rely on a book of that age for a definition of any entity in modern mathematics. With that definition, as an example, you would have to know how to count the number of things (in a mathematical precise way), which will get you to the next lacking definition. Today, most of basic mathematics relies on set ...

3

As is often the case, Wikipedia is your friend: see Set-theoretic definition of natural numbers. Briefly: $0$ is defined as the empty set $\emptyset$; and $n+1$ is defined as $n \cup \{n\}$.

3

Suppose that $\overline{aA_+a}\subsetneq A_+$. Use Hanh-Banach to construct a functional with $f(b)=1$ and $f(aca)=0$ for all $c\in A_+$. This functional, extended to all of $A$, is a linear combination of four states (see II.6.3.4 in Blackadar): at least one of them, say $f_1$, will satisfy that $f_1(b)>0$; by the Jordan Decomposition, we also have that ...

3

No, the tensor product isn't zero. Be more careful about the definition. The way tensor product is constructed is to take the free group generated by elements of $A \times B$ and what we really want is to have identities such as $(a+a',b)=(a,b)+(a',b)$ so we take the subgroup generated by those three above types of elements and then taking quotient would ...

3

First assume: $\varepsilon>0$, there exists a corresponding number $\delta>0$ such that: $$\text{If } 0<|x-a|<\delta \text{ then } |f(x)-L|<\varepsilon$$ For that definition take $\varepsilon=\frac{\epsilon}{2}$, then exist $\delta$ such: $$\text{If } 0<|x-a|<\delta \text{ then } |f(x)-L|<\frac{\epsilon}{2}$$ Then you can see: ...

3

Here is a very, very quick answer that should help you: if you want $\leq \varepsilon$, just choose $2\varepsilon$ in the definition with $<$; if you want $\leq \delta$, just choose $\frac{\delta}{2}$ in the definition with $<$. In the first case, you exploit the words "for all", in the second the words "for some".

3

The two definitions are equivalent, let me give a quick proof: Let $B\subseteq E$ be a subspace of a metric space $(E,d)$. $\Rightarrow$: Assume there is an $r>0$ such that $d(x,y)<r$ for all $x,y\in B$. Fix any $p$ in $B$ then $d(p,x)<r$ for all $x\in B$. $\Leftarrow:$ Assume there is a $p\in E$ and $r>0$ such that $d(p,x)<r$ for all $x\in ... 3 To be honest "incomple information" is not a good terminology. If players do not know all the relevant information, the game is not really well specified. But for historical reasons, the term has stuck. When you want to model a situation of "incomplete information" what you do in practice is to use Harsanyi's trick: you replace the incomplete information ... 3 Here's another attempt at doing the whole thing, without the possibly confusing notation. Let's define a kind of object that we'll call a "prn". A "prn" is defined as an ordered pair$(a, b)$(with some additional structure that we'll define below), and to make it clear that it's a "prn" and not just an ordered pair, we'll denote$(a \star b)$. We can ... 3 We say that an implication$p\to q$holds vacuously if$p$is always false. That is to say, it is impossible to have$p$true and$q$false. So the implication is a tautology. Of course tautologies exist in propositional calculus, and not quite in predicate logic (and thus not in first-order logic), but the concept caries over. So when we say that the ... 3 You are "not alone" with your doubt about$\emptyset$; see the "debate" in this post. You must "work with" Asaf's answer: basically, we have the definition of$\emptyset$and that of inclusion :$A \subseteq B =_{def} \forall x (x \in A \rightarrow x \in B)$. We have also a "basic principle" of mathematical reasoning (but not only) : "stay with the ... 3 We all agree about the "arbitrary" nature of Kuratowski's definition, and about the possibility of replacing it with other "conventions" (equally arbitrary), provided that they satisfy the basic properties (already discussed). But please, note that the lack of a mathematical definition of ordered couple forced two of the "founding fathers" of modern ... 3 The definition of odd and even function requires the domain$D_f$of$f$be such that$\forall x\in D_f\left(-x\in D_f\right)$. When$D_f$satisfies this property, then$f$is said to be odd if$\forall x\in D_f(f(-x)=-f(x))$and it is said to be even if$\forall x\in D_f(f(-x)=f(x))$. And also, regarding a problem such as this, can anybody tell me if ... 3 In mathematics definitions (and axioms) are the attempts to formalize some informal notions. Sets come to formalize the notion of a "collection", so we can talk mathematically about collections of objects. The collection makes a distinction between two things which are not equal, but that's it. So if I open my wallet, and look at my coins, while I might ... 2 A basis is any set of vectors that are both linearly independent and span the space. If$V$is a vector space and$b_1, ...,b_n$are a basis of$V$this means that (1) any$v \in V$can be written in the form$c_1b_1 + ...+c_nb_n$where$c_i \in \mathbb R$(that is saying that they span the space) and (2) if$c_i$are such that$c_1b_1 + ...+c_nb_n = 0$... 2 The defining property of a basis is that every vector in the vector space can be written as a unique combination of the basis vectors. This is written as, $$\vec{v} = a_1 \vec{e}_1 + a_2 \vec{e}_2 + \cdots + a_n \vec{e}_n.$$ With the understanding that this can be done for every$\vec{v}$in the vector space. The coefficients$a_k$are the components of ... 2 This is an informal answer, but hopefully'll give you an idea of the concept. Let's pick an easy vector space,$R^2$(which is just the familiar xy plane). Hopefully that makes sense to you that this is a vector space. If you pick any 2 points on the plane (that aren't "in line" with each other from the origin), can you see how linear combinations of ... 2 This is a really good question and it comes up (among other places) in tensor category theory, where equality is replaced by canonical isomorphism and suddenly things cannot be handwaved away... I'll try to give an answer; unfortunately it is lacking in illustrations (due to math.stackexchange not having packages like xypic) and examples (due to my ... 2 One can, but I do not think one should. Also,$n$-tuples are not basic objects of standard set theories, they have to be defined. Take a probability space$(\Omega,\Sigma,\mu)$. Should we take this as$(\Omega,(\Sigma,\nu))$with some definition of an ordered pair? How about$((\Omega,\Sigma),\mu)$? Or maybe it is a function$f$with domain$\{0,1,2\}$... 2 It's often okay to define a mathematical object without an explicit set theoretic model. If we need a model we can always take a step back and make sure there is one. Take for example the following definition: A map$f$from a set$A$to a set$B$assigns to every$a\in A$a unique value$f(a)\in B$. We write$f\colon A\to B$. Two maps$f,g\colon A\to ...

2

A precise formal definition of a (winning) strategy for Duplicator really is not very enlightening. A strategy would tell how Duplicator should play given the position of the current play of the game. Each play of the game is essentially a sequence in $A \cup B$:  \begin{array}{r|cccccccc} \text{Spoiler} & c_0 & & c_2 & & c_4 ...

2

The first thing to do is to compute some of the transition maps. A point where the Mobius band twists has a transition map that's multiplication by a negative element of $\mathbb{R}$. To see this, just write down a trivialization via two charts. This leads to the fact that a bundle is trivial, i.e. homeomorphic to the Cartesian product of the base and the ...

2

Formally speaking, the series is divergent. The sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. We see that the sequence of partial sums of the series, known as Grandi's series is $1, 0, 1, 0, …$, which does not approach any number (although has two accumulation points at $0, 1$). Therefore, we may ...

2

Posets whose all non-empty subsets have an infimum and a supremum inside are exactly the finite totally ordered sets. The name of such a poset is "finite chain". Indeed, let $(D, \leq)$ be a such poset. First, it is totally ordered: for any $x, y \in D$, consider the subset $\{ x, y \}$. Now, since all non-empty subsets have an infimum inside, $(D, \leq)$ ...

2

That's a question. Well for start as shown in Mac Lane Category theory for working mathematicians there are two different way to approach categories, functor and natural transformations: you can either reguard categories as some family of sets and operation between them (eventually adding some axioms to set theory since you would like to work with large ...

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