I have seen the definition of a set which states it is a collection of well defined distinct objects. But there is some ambiguity for the words well defined and objects. I want to know is there a concrete definition for a set? Any further idea or reference will be very helpful. Thank you.
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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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I am familiar with the set-theoretical definition of functions.   A function $f : A \rightarrow B$ from a set $A$ to a set $B$ is a subset $f \subseteq A \times B$ of the Cartesian product $A \times B$ such that the following conditions hold:   $\forall a \in A : \exists b \in B : (a,b) \in f$ $\forall a \in A : \forall b, b' \in B : (a,b), (a,b') \in f \rightarrow b=b'$   Is there a purely set-theoretical definition of the function evaluation $f(a)$ for every $a \in A$? I get as far as defining  $f(a) = \{ b \in B : (a,b) \in f \}$ but that defines rather the set of values of a multi-valued function.
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Can someone give an argument, if possible using only the axioms of set theory, because I'm very weak there and have virtually no background, except the usual knowledge of the operation with sets one has to have when doing non-set theoretic non-research mathematics, why $\emptyset \in \emptyset$ or $\emptyset \subseteq \emptyset$ should or should not hold?
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I have came across this in a textbook:     $\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$   I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset?
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Suppose that a set has an odd number of elements. Explain why half of the subsets will have an odd number of elements .  Now assuming set A is the set with an odd number of integer elements{1,2,3,4,5}  Subset b ={1,2}  subset c={3}  Subset d={4,5}  Now there are two subsets with even number of elements and one subset with an odd number of elements.It seems to contradict the earlier theorem that i have to prove.I think that i am missing something.
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Is "empty set" an element of a set? For example:  is this FALSE?:  0 ∈ {1,2,3}
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I was wondering if there is a reason why in the definition of a topologic space, the reunion of any collection of open sets is also an open set but the intersection has to be finite to be a set.  Is there because of the definition of an open set itself ? I have always seen it as a definition, but now I want to know why.  Thanks for your help.
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We define the term "Set" as,     A set is a collection of objects.   And an "Empty set" as,     An empty set is a set which contains nothing.    First problem I encountered:  How the definition of "Empty set" is consistent with the definition of "sets" if "Empty set" contains nothing and a "set" is a collection of objects.  Further we discovered in set theory that every set has a subset that is the Null set.  Such as,  If $A=\emptyset$ and $B=\{1,2,3\}$ then,  $$A \subset B$$  Second problem I encountered:   How and why "No element" is referred and considered as an element as we do in case of null set that is, when we say that every set has a subset that is the Null set?  Third and Last one:  How can a set possess "some thing" and "nothing" simultaneously that is when we say that every set (containing objects) has a subset which is Null set (contains nothing)?
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I've been trying to challenge myself but with no luck. Maybe one of you will have a better idea. How can it be proven that there can't be a set which contains its own powerset, using only russell's paradox? I've managed to prove it with other principles or axioms, such as that a set can not belong to itself, or that a powerset cannot belong to itself.
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Let F be a set of sets. We say a set is n-pretty  (it's writen in my text book, I don't know the exact name)(n is fixed nat. number) if $$\forall X (X \in F \iff \forall Y (Y \subseteq X \land |Y| &lt;= n \implies Y \in F)$$. And we say that a set is finite-pretty if: $$\forall X (X \in F \iff \forall Y (Fin(Y) \land Y \subseteq X \implies Y \in F)$$ So the task is to find if every n-pretty set is finite-pretty and if every finite-pretty set is n-pretty. I think that every finite-pretty set is n-pretty. Since it's true for every finite set, it will be true for every set with cardinality n. But I can't seem to  prove it in terms of set theory. And what about the other direction ? Any tips?
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In ZFC, for example, there is no universal set, so what does it mean to write $\forall x (\cdots)$, i.e., for all sets something is true? Does it avoid the problem by quantifying over all elements but not allowing those elements to form a set?  A related question I have is about quantifying over infinite sets. If $X$ is a finite set with $n$ elements $x_1, x_2, x_3, \ldots, x_n$, Then it is reasonable to define $\forall x\in X :P(x)$ to be the statement $P(x_1)\wedge P(x_2)\wedge\cdots\wedge P(x_n)$ (which can be formalized using a recursive definition). But what about infinite sets? It seems that if $Y$ is a countably infinite set with elements $y_1,y_2,\ldots$, then $\forall y\in Y: P(y)$ would mean $P(y_1)\wedge P(y_2)\wedge\cdots$, which we couldn't finish stating. For uncountable sets, we would miss an element in between every time we count upwards.
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Hello I am taking a Discrete Mathematics course and having some trouble with this question about sets:  Under what conditions is $A- B = B- A$  Diagram:   Diagram:   Maybe I'm understanding incorrectly, but how can A-B and B-A be equal if they contain elements that are not in each other?
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We define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$.  How can we prove that a subset of a finite set is finite? It is of course sufficient to show that for a subset of $\{0,\ldots,n-1\}$. But how do I do that?
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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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Recently I learned that for any set A, we have $\varnothing\subset A$.  I found some explanation of why it holds.     $\varnothing\subset A$ means "for every object $x$, if $x$ belongs to the empty set, then $x$ also belongs to the set A". This is a vacuous truth, because the antecedent ($x$ belongs to the empty set) could never be true, so the conclusion always holds ($x$ also belongs to the set A). So $\varnothing\subset A$ holds.   What confused me was that, the following expression was also a vacuous truth.     For every object  $x$, if $x$ belongs to the empty set, then $x$ doesn't belong to the set A.   According to the definition of the vacuous truth, the conclusion ($x$ doesn't belong to the set A) holds, so $\varnothing\not\subset A$ would be true, too.  Which one is correct? Or is it just a convention to let $\varnothing\subset A$?
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I Have a question. Given only the definition of equality of two sets, how can we prove that there is one and only one empty set. I mean by equality of two sets the following: $$A=B \iff  \forall x (x \in A \iff x \in B)$$