I am curious of that words; 'family' and 'set.  Some use the set of the blah-blah-blah~.  However, the other some use the family of the blah-blah-blah~.  For example, we express $X=\{v_i\}_{i=1}^N$ as   $X$ is a set of $N$ vectors $v_1, v_2, \ldots, v_N$. $X$ is a family of $N$ vectors $v_1, v_2, \ldots, v_N$.   How can I understand the diffence between them?
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What is the difference between "family" and "set"?  The definition of "family" on mathworld () is a collection of objects of the form $\{a_i\}_{i \in I}$, where $I$ is an index set. But, I think a set can also be represented in this form. So, what is the difference between the concept family and the concept set? Is there any example of a collection of objects that is a family, but not a set, or reversely?  Many thanks!
97767
I can't fully understand the general definition of the Cartesian products of an arbitrary collection of sets defined by an index set. The general definition states that it is simply the set of all functions from the index set to the union of the sets in the collection. Now since functions are formally defined as ordered pairs, how can the set of all functions generate a set of n-tuples, given that the collection contains n sets?  Thanks.
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I got the answer as the set $\{5x + 1 : x \in \mathbb{N} ∪ \{0\} \}$. Can anyone confirm if this is correct? Thanks.
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Take for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?
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Does a family of a distribution have a different definition for statistics than in other disciplines?  In general, a  is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is varied. Such families are used, for example, to .  For statistics, a family according to  is the result of varying the shape parameter. How then can we understand that the gamma distribution has a shape and scale parameter and only the generalized gamma distribution has, in addition, a location parameter? Does that make the family the result of varying the location parameter? According to @whuber the meaning of a family is implicitly A "parameterization" of a family is a continuous map from a subset of ℝ$^n$, with its usual topology, into the space of distributions, whose image is that family.  What, in simple language, is a family for statistical distributions?  A question about relations among of the statistical properties of distributions from the same family has already generated considerable controversy for a  so it seems worthwhile to explore the meaning.  That this is not necessarily a simple question is born out by its use in the phrase , which has nothing to do with a family of curves, but is related to changing the form of the PDF of a distribution by reparameterization not only of parameters, but also substitution of functions of independent random variables.
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I don't understand why a set with n elements has 2^n subsets. How is this calculated? I realize that {123} has empty set - 1-2-3-1,2-1,3-2,3-1,2,3 but how is the formula derived?
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From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not .  My question is what the definition of a set is?   I have noticed that many other definitions start with a set and then something. A  with an operation, an equivalence relation is a set, a , even the  of other sets containing the empty set.  I understand that there is a whole area of mathematics (and philosophy?) that deals with . I have looked at a book about this and I understand next to nothing.  From what little I can get, it seems a sets are "anything" that satisfies the axioms of set theory. It isn't enough to just say that a set is any collection of elements because of various paradoxes. So is it, for example, a right definition to say that a set is anything that satisfies the ZFC list of axioms?
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I am having difficulty understanding  :     Every non-empty set $\rm A$ contains an element $\rm B$ which is disjoint from $\rm A.$   So from my understanding if I have a set like:   $$\{1, 2, 3, 4, 5\}$$  Then one of $[1,2,3,4,5]$ is not an element in $\rm A?$ huh?
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Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty set, have infitinite emptiness e.g. do all sets including the empty set contain infinitely many empty sets?
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This is one question which has been asked in Velleman's How to prove book:     Suppose $A$ is a set, and for every family of sets $F$, if $\cup F = A$ then $A ∈ F$. Prove that $A$ has exactly one element   This question has been already proved  but isn't something like this an counterexample of the proof:  $ F = \{\{1\},\{2\},\{1,2\}\}$   $ A = \{1,2\} $        Now,  $ \cup F = \{1,2\}$  and also $A \in F$ but $A$ has more than one element.             Is my counterexample valid ?
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Let $N$ and $N'$ be two sets. Is it possible that $N \subseteq N \times N'$?  I don't think so, but actually I am not able to get any absurd sentences supposing that is true.   Let us suppose that $N \subseteq N \times N'$. Then, for any $x \in N$ there exist $n \in N$, $n' \in N'$, such that $x = (n,n')$. But then there exist $m \in N$, $m' \in N'$, such that $n=(m,m')$, $x=((m,m'),n)$ and so on. I see it is ugly, but I can't understand why it is absurd.
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I have alot of questions.   Do Infinite sets have the same cardinality when there is a bijection between them? Are $\mathbb{N}$ and $\mathbb{Z}$ infinite sets? I assume they are, but why? Why does there not exist a bijection between them? Why do $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ have the same cardinality? I assume there exists a bijection between them, but how can I find this function?
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If the empty set is a subset of every set, why it isn't written with the elements of a set? like so $\{1,2,3,\emptyset\}$   Or why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements and the other has 1 but since the empty set is a subset of both, then why it isn't being mentioned explicitly in the definition of the set?
5263
Let $ D = 2^\mathbb{N} $, i.e., D is the set of all sets of natural numbers.   What's the meaning of this definition? Intuitively, I would suggest that $ D = \{1,2,4,...\} $ but the explanation "set of all sets" leads me to the guess that this is wrong.
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I am not very well versed on set theory or syntax, but I thought I knew the basics.  However, in a book about databases I am reading now, the author uses $2^x$ to signify "a set of $x$."  For example, $2^{\text{dogs}}$ is a set of $\text {dogs}$, etc.    The author never really explained this or why he does it, I just picked up the meaning from context.    I am not sure why the exponent operator is used, nor am I sure what the number $2$ has to do with it.  The sets being represented are NOT powers of $2$ (in size)... they come in all sizes.  Is this a valid notation?  I have not seen it anywhere before...
118987
I'm reading Guerino Mazzolla's Comprehensive Mathematics for Computer Scientists 1:     Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set      $$\{x| \text{ there exists an element } b \in a \text{ such that } x\in b \}$$      This set is denoted by $\cup a$ and is called the union of $a$.    This axiom seems to take elements from inside of a set of sets, for example in the set: $a=\{\{u_1,u_2 \},\{u_3,u_4 \} \}$, there are two elements $b\in a$, then the mentioned set created by the axiom is $\{u_1,u_2,u_3,u_4 \}$.           The axiom seems to imply that the set $a$ must have a set inside of it for it to work, what if it  have an element that is not a set? (I'm not sure if this is possible, but I remember of having read something about the difference of sets and elements).         There's also an extension of my proposal, In the same book (and also in others) I've seen that the natural numbers are defined as:   $$\begin{eqnarray*}   {0}&amp;=&amp;{\emptyset} \\    {1}&amp;=&amp;{\{\emptyset\}} \\    {2}&amp;=&amp;{\{ \{\emptyset\}\}} \\    {3}&amp;=&amp;{\{ \{ \{\emptyset\}\}\}} \\   {\vdots}&amp;=&amp;{\vdots} \end{eqnarray*}$$  Then the singleton $\{ 3\}$ could be seen as $\{ \{ \{ \{\emptyset\}\}\}\}$, which could mean that   $$\cup \{ 3 \}=\cup\{ \{ \{ \{\emptyset\}\}\}\}= \{ \{ \{\emptyset\}\}\}=\{2 \}$$          Then the union could also be seen as the antecessor of a number?            Using the framework I provided, I see some ambiguity, look:   $$\{ \{ \{\emptyset\}\}\}=\{2 \}$$          I decided to interpretate it this way, but from the notation it's not clear in which pair of brackets I should stop, I guess that $\{ \{ \{\emptyset\}\}\}=\{2 \}$ but it could also be the case that $\{ \{ \{\emptyset\}\}\}=3$. I could want a singleton with a $2$ inside of it and a $3$ outside a set, no?
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A question from my homework I'm having trouble understanding. We are given:  $A(1) = \{\varnothing\}$, $A(n+1) = A(n)\cup (A(n)\times A(n))$  $A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots$ to infinity  The questions are:  1) show that $A\times A \subseteq A$  2) Is $A \times A = A$?  Thank you for your help.  I've tried writing $A(2)$ but it gets really complicated and I'm having trouble understanding what the sets are. Let alone solve the question.