Set theory Definition and 439 Threads

It has been known for some time that the Axiom of Choice (if you treat it as a proposition to be proved rather than an axiom) and the Continuum Hypothesis are independent of Zermelo-Fraenkel set theory (ZF). These and other statements (Suslin's Problem, Whitehead's Problem, the existence of...

Homework Statement I have been asked to show that \exists xAP(x)\vee\exists xBP(x) is equivalent to \exists x(A\cup B)P(x) Homework Equations 1) P\rightarrow Q \equiv \neg P \vee Q 2) \neg(P\vee Q)\equiv \neg P \wedge \neg Q 3) P \vee (Q\vee R) \equiv (P\vee Q) \vee R \equiv P...

Im just going through a practice exam and I was wondering If I could get someone to check my results. Q1 Let X = \{1,2,4,5\}, where elements of X are the sets 1 = \{0\}, 2 = 1 \cup \{1\}, etc. Evaluate each of the following: a) X \backslash \{2\} = \{1,2,4,5\} \backslash \{2\} = \{1,4,5\} b)...

If P(X) denotes the power set of X. Is |P(A)| = |P(B)| iff A=B true? If so, I have no idea how to prove the |P(A)| => |P(B)| iff A=B direction, so any hints would be great.

Hello everyone. Our book has a problem where we are to fill out the missing spots and its quite confusing, I'm not sure if i got this right or not. Any help would be great! Here is the question/directions: The following is a proof that for all sets A and B, if A is a subset of B, then A U B...

Hello everyone. I think i have this right but im' not 100% sure due to the last set, A_3 Here is the problem: http://suprfile.com/src/1/3m5fyjg/lastscan.jpg Here is my answer: Yes. By the quotient-remainder theorem, every integer n can be represented in exactly one of the three forms. n =...

Let F be the label of an non-empty set and let (B_m)_{m \geq 1} be elements in 2^F Then I need to prove the following: \mathrm{lim}_{m} \ \mathrm{sup} \ B_{m} = \mathrm{lim}_{m} \ \mathrm{inf} \ \mathrm{B_m} = \cup _{m= 1} ^{\infty} B_{m} if B_{m} \uparrow which implies that B_{m}...

If U [i.e., set theory] were to be equippable with a vector space type morphology...Prolly more of a module than a v.s.. Yes, a field over a ring, perhaps, if that's possible... dim(U)...: 0. emptiness 1. isolation 2. expansion 3. containment 4. transition 5. hyperspace 6...

This year I’m going to be a senior and I’m going to start applying to graduate programs soon. I’m a pure math major with a philosophy minor. I really would like to do my grad work in set theory, or become a logician. But the career prospects look grim in these fields. Is this the case or am...

I want to make sure I understand the meaning of membership and subset. For example, if I have a set x, then is x a member/subset of the set S = {{y},x} I came to the conclusion that x is a member of the set S because S contains x as an element, and x is also a subset of S because S...

The question is If A U B is a subset of A intersect B, then prove that A=B Now i can see this in my head and it makes sense that the elements in set A and Set B would have to be the same. The problem that i have with subset questions is how to prove that this is the case. I can start by...

It makes me wonder what should be studied first - whether the basics of axiomatic set theory or mathematical logic? Although I initially that logic should be studied first, set theory second, now something makes me think that it should be vice-versa. The reason for this shift is that - when...

I have some understanding problems with what the prof taught me today. I am just going to break it down and we can discuss, perhaps: a. the sum of the collectively exhaustive events must equal 1. I know that if an event is both collectively exhaustive and mutually exclusive it should cover...

1)let P={p1,p2...} be the set of all prime numbers for every n natural, present n as a representation of its prime factors, n=product(p_k^a_k) 1<=k< \infty (where a_k=0 besdies to a finite number of ks). and now define F:N->Q+ by: F:n=product(p_k^a_k)->r=procudt(p_k^f(a_k)), and show that's a...

What/where is the current research being done in set theory? I’ve been looking into grad programs but I can’t seem to find any that allow for set theory as an area of research. Does anyone know of any (preferably in California)?

I am having a nightmare trying to prove things in set theory. One of my homework problems is to prove that: Dom(R U S) = Dom(R) U Dom(S) but i have no idea how to really do this. my teacher never went over this stuff! IT'S SO AGGRAVATING! can anyone reference a good site or book on...

i need to prove that next three arguments are equivalent: 1)f:X->Y is on Y. 2) f:p(X)->p(Y) is on p(Y). 3)f^-1:p(Y)->p(X) is one-to-one correspondence. where p is the power set.

Let A and B be subsets if a universal set U. Prove the following. a) A\B = (U\B)\(U\A) To do this, show it both ways. 1) A\B contains (U\B)\(U\A) 2) (U\B)\(U\A) contains A\B I'll start with 2) if x is in (U\B)\(U\A), then x is in (U\B) and x is NOT in (U\A). then (x is in U and x...

Does anybody in here know their Set Theory really well? I could do with some help on a few questions! Q1) Show how an equilance relation on a set X leads to a partition of X? Q2) Let A and B be sets and f: A \rightarrow B be a function. For each b \epsilon ran f. Show that the collection...

If E has m elements and F has n elements, how many elements does E x F have? My thinking is that E x F would either have m or n elements. If m= n, then E x F would have m elements (or n elements). If m>n, then E x F would have n elements since E x F ={(x,y): x is an element of E and y is an...

i searched in the homework section and there isn't a section for logic an set theory so i ask my questions here (begging for replies): 1)expand the proposition (by the equivalence rules): [~(pvq)v((~p)^q)] i got to this: [(~pv~p)^(~pv~q)]^[(qv~p)^(qv~q)] is it correct? 2) prove/disprove...

I'm about to read "Naive Set Theory" by Paul R. Halmos. Amazon sells one published by Springer (1st edition, 1998) while my library (Universitas Gadjah Mada, Indonesia) has one published by Princeton (1st edition, 1960). Is the content any different? If it is significantly different I'll try...

As I have not received an answer for the post I submitted under the thread "Does the set theory prove that there is no God?" I decided to submit it here. Without having much knowledge of set theory, How exactly does the Set theory prove that there is no god? Thank you in advance for your kind...

Can someone introduce me to what the basics of set theory are, mean, are use in, have been created by, and it's branches?

One of the axioms of the axiomatic set theory is that there are no universal sets. However, God is omnipresent, so he would have to be this universal set. Thus, God would immediately lead to a contradiction within the set theory. Does this prove that there is no God?

Dear members, it has recently been put forth to me the question as to when and by which means did it become clear to the majority of logicians and mathematicians that set theory prevailed over Russell's theory of types (as in Principia)? (By set theory I mean as in the axioms of...

Hi...I'm wondering if infinite set theory has any place in finding limits. Is there any way that tabling elements of sets can find you the answer to a limit question?

Hi all (i) A- (B-C) = (A-B) U C (ii) A - (B U C) = (A-B) - C Which one is always right and which is sometimes wrong? My solution If x is an element of A - (B-C), then x is not contained in B-C. If y is an element of (A-B) U C, then y is at least one of A-B or C. (i) is sometimes...

Hello all Set X has x elements and Sset Y has y elements and Set Z consists of all elements are are in either set X or set Y with the exception of the k common elements. I know that the answer is x+ y - 2k, however how would I get this? Should I just use a practical example?

What are the weaknesses and strengths of Russell's type theory? I'm having trouble understanding exactly what makes it weaker than ZF. I've been told its too restrictive, but what exactly makes it more restrictive than ZF? Are there any objections to ZF?

I have a question about the potentially self-referential nature of cantor's diagonal argument (putting this under set theory because of how it relates to the axiom of choice). If we go along the denumerably infinite list of real numbers which theoretically exists for the sake of the example...

I am currently reviewing for an upcoming test over sets. What the instructor did was to give out the test he gave out for last semester for us to study from. I can answer most of these questions but there are a few that I am a little bit unsure of. Some of the questions are complete the...

Let C be a circle and let D be the set of all diameters of C. What is \capD? I think it is the center of the circle since that would be the only point of intersection of all the diameters of the circle. Could someone let me know if I am correct? Regards Jeremy

ok, find the intersection of i=1 to infinitie of A(i). A(i) = [0, to 1/i]. I don't understand what it is asking. what do they mean to find the intersection of that? and what if A(i) = [0, 1/n)

I'm trying to prove something small with set theory and since I'm new to it, I've run into a problem. I can't understand what the following means exactly and how to proceed further. Or where the mistake is, if there is one. I think there is, because it seems... freaky...

Let f:X->Y be a function 1) Given any subset B of Y, prove that f(f^-1(B)) is a subset of B 2) Prove that f(f^-1(B))=B for all subsets B of Y if and only if f is surjective Help anybody?

I have to prove the following theorem, 1) If f:A->B is a surjection, and g:B->C is a surjection then g dot f:a->C is a surjection Well this makes sense and I am not sure how to PROVE it Is it sufficient to say the following if for every element b of B, there exists a element a of A...

Let f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S = {f(c): 0<c<1}? I. S is a connected subset of R II. S is an open subset of R III. S is a bounded subset of R The answer is I and III only. I understand...

was there an attempt to unite between those two fields?