Cardinality Definition and 167 Threads

Homework Statement Find the number of elements in the ring Z_5[x]/I, where I is a) the ideal generated by x^4+4, and b) where I is the ideal generated by x^4+4 and x^2+3x+1. Homework Equations Can't think of any. The Attempt at a Solution I started by finding the zeros of the...

Homework Statement Do the following have the same cardinality? If so, establish a bijection and if not explain why. A line segment of 4 units and half of a circumference of radius 1 (including both endpoints). The attempt at a solution So my thought is that if I manipulate the...

Under the continuum hypothesis, we readily see that that |{a < \aleph_1 : \textrm{a is a cardinal}}| = \aleph_0 . What happens under the negation of CH? Is this equality still true or not? If the latter, always under the negation of CH, are there any infinite cardinals lambda for which the...

Hello, given a set \Omega, we consider all its subsets A_1,A_2,A_3,\ldots with same cardinality k. Do you have some hint in order to prove the following: \forall A_x,A_y,A_z\subseteq \Omega such that |A_x|=|A_y|=|A_z|=k |A_x-A_z| \leq |A_x-A_y|+ |A_y-A_z| Thanks

Is Card (N even)< Card (N)? Where N even is set of all even Natural numbers, N is set of all Natural numbers. Hint: use the mapping from N eve→Nn N even to N is given by n-->n a. Show examples of this mapping from N even N even to N. b. Is the mapping above onto? One-to-one? My try...

For any two rational numbers q1<q2, card ((q1,q2) cap Q) = card Q, right? How to prove it, if it's true?

Homework Statement Every subset of \mathbb{N} is either finite or has the same cardinality as \mathbb{N} Homework Equations N/A The Attempt at a Solution Let A \subseteq \mathbb{N} and A not be finite. \mathbb{N} is countable, trivially, which means there is a bijective...

Hey guys, this is my first post, (Hi) was just wondering if i could get your help. I'm studying for my repeats and you guys can save me. If X = {1,2,3,4}, Y = {2,4,6} what is the cardinality of the following sets? (i) A = {x|x mod 2 = 0 and 0 <=x<=20} (ii) B = X * X * Y (iii) C = {(x,y)|x ≠ y...

Homework Statement Let S be the set consisting of all sequences of 0's and 1's. S = {(a_1, a_2, a_3,...):a_n = 0 or 1 } Show that S is uncountable. Homework Equations The Attempt at a Solution I'm not sure where to start. I think I should either assume that S~N and use...

What's the cardinality of \mathbb{R}^{\mathbb{N}}? It must be \mathbb{R} or \{0,1\}^{\mathbb{R}}, but I'm not sure which.

A=\mathbb{R}-0 (0,1)\subseteq \mathbb{R}-0 Assume A is countable. Since A is countable, then A\sim\mathbb{N}. Which follows that (0,1)\sim\mathbb{N}. However, (0,1) is uncountable so by contradiction, Card(A)=c Correct?

Prove cardinality of every finite nonempty set A is less then cardinality of natural number N |A|<|N| set A is nonempty finite set natural number N is denumerable (infinite countable set) |A|<|N| if there exist a injective (one-to-one) function f: A->N, but NO bijective function, which...

If C is countable, then |N|=|C|, so there exists a bijection bewteen N and C, but does this imply that |R\N| = |R\C|? It is intutively believable, but I don't see how it rigorously follows. What is the bijection between R\N and R\C? Is it provable in general that If |A|=|C| and |B|=|D|...

Homework Statement Q1) Assuming that |R|=|[0,1]| is true, how can we rigorously prove that |R2|=|[0,1] x [0,1]|? How to define the bijection? [Q1 is solved, please see Q2] Q2) Prove that |[0,1] x [0,1]| ≤ |[0,1]| Proof: Represent points in [0,1] x [0,1] as infinite decimals...

Homework Statement Assuming the fact that the set of algebraic numbers is countable, prove that the set of transcendental numbers has the same cardinality R, the set of real numbers. Homework Equations N/A The Attempt at a Solution Let A be the set of algebraic numbers and T the set of...

Would the cardinality of a set S = {a, b, {c, d}} be 5?

Homework Statement What is the cardinality of the set of all continuous real valued functions [0,1] \rightarrow R . The Attempt at a Solution In words: I will be using the Cantor Bernstien theorem. First the above set, let's call it A, is lesser then or equal to the set of all...

If S1 and S2 are finite sets, show that |S1 x S2| = |S1||S2|. Here is what I've tried: Let |S1| = m and |S2| = n. Let P(k) be true. That is, P(k) = |S1 x S2| = km. P(1) is true since, if |S1| = 1 and |S2| = 1, |S1 x S2| = 1. Now, let |S1| = k+1 and |S2| = m. Then, P(k+1) = |S1 x...

I need help proving this: Find an explicit one to one correspondence between the interval (-1;7) and the real numbers R Any ideas?

Need help with this proof: Prove that the set of irrational numbers has the same cardinality as the set of real numbers.

There is an argument to account for the Casimir effect based on cardinalities: that inside the two plates only (virtual) photons with wavelengths corresponding to the harmonic series can exist, hence countable infinity, whereas photons of all wavelengths can exist in the space around it, hence...

I was reading Hodges' Model Theory when I came across this question in the first chapter: Specify a structure of cardinality w2 which has a substructure of cardinality w but no substructure of cardinality w1. (Working in ZFC) I am assuming w2 means 2w1 but I'm not sure. I haven't really...

1. If X is an infinite set and x is in X, show that X ~ X \ {x} A~B if there exists a one-to-one function from A onto B. Attempt at a solution I'm pretty much completely stumped on this problem. I know that since X is infinite then it contains a sequence of distinct points. So...

Homework Statement Prove the set of continuous functions from R to R has the same cardinality as RHomework Equations We haven't done anything with cardinal numbers (and we won't), so my only tools are the definition of cardinality and the Schroeder-Bernstein theorem and its consequences. I...

Can anyone please give a really explicit proof (omitting no steps) and with as simple words as possible that any infinite set can be writtern as the union of disjoint countable sets? Thank you.

A set of cardinality \aleph_0 has elements that are sets of size \aleph_0, and so on. Counting elements, I get \aleph_0 + \aleph_0^2 + ... + \aleph_0^{\aleph_0}\ . Is this the same size as \aleph_0 ?

If I know: |A \cup B \cup C| = 1000 |A| = 344 |B| = 572 |C| = 296 |A \cap B| = 301 |B \cap C| = 252 |A \cap C| = 213 and I use the standard formula to compute |A \cap B \cap C|, I get 554, which is absurd. Can someone tell me what's wrong here? Is there something inconsistent in the initial...

Show that the Cardinality of a set which doesn't include inverse elements from a group G is always even. So the set with all g in G, which includes no inverse elements from G. (g=!g^-1) I can get this in every example I've done, checking mainly with dihedral groups, it's always been true but I...

The cardinality of set of [N]\omega . what does omega stands for?

The cardinality of set of [N]^\omega . what does omega stands for?

Since Tex is giving me a hard time this question is going to be a lot more undeveloped than it was going to be. We were discussing in class the cardinality of the Natural numbers being aleph null and that the cross product was also of cardinality aleph null. Is it true then that...

Homework Statement Find the cardinality of the set of all equivalence relations on N Homework Equations by what we have learned yet we only have to determine if it's countable or not The Attempt at a Solution I know that the set of all relations on N is equivalent to P(NXN) thus is...

Homework Statement Prove the intervals of real numbers (1,3) and (5,15) have the same cardinality by finding an appropriate bijective function of f:(1,3) ->(5,15) and verifying it is 1-1 and onto Homework Equations I know there are multiple ways to prove one to one and onto I am not sure...

what is the cardinality of a set A of real periodic functions ? f(x)=x is periodic so R is subset of A but not equal because sin(x) is in A but not in R. hence aleph_1<|A|.

Homework Statement Let a,b\in\Re such that a < b. Show |(a,b)|=|(0,1)| and Show |(0,1)|=|(0,\infty)| Homework Equations The Attempt at a Solution] I know that |(0,1)| is equal to the continuum and has the same cardinality as the reals. I would guess that |(0,\infty)| is also...

I was reading about this topic of my own leisure, and I came across something that I couldn't quite understand. The solution of Galileo's Paradox is that the set of natural numbers and the set of perfect squares are both infinite sets of the same cardinality (namely aleph 0). This I can...

How do I prove that the set of all finite subsets of an infinite set has the same cardinality as that infinite set?

Something has been bothering me. I can't sleep, and neither can the little people inside my head. I'm not a mathematician, and I don't understand what I'm dealing with all that well, but I'm trying to figure out if the following makes sense: The cardinality of R^{1} is equal to the...

Can you prove the following theory of cardinality for a Cartesian product, - \left|\:A\:\right|\:\leq\:\left|\:A\:\times\:B\:\right|\: if\: B\neq\phi In English, The cardinality of a set A is less than or equal to the cardinality of Cartesian product of A and a non empty set B.

Is it true that the set of permutations on a infinite set K has the same cardinality as all functions between a infinite set K to itself?

Hi there, I'm having a lot of trouble understanding this particular problem, and I hope the fine people of this forum can help me out =) Homework Statement Let A and B be infinite sets with the same cardinality. Prove that P(A) and P(B) have the same cardinality. Do this by giving...

Homework Statement 1) Find the cardnality of the set of constructible angles whose cosine is a) irrational b) rational 2. The attempt at a solution For the other questions I have posted my attempt, but for this one I really have no clue... Could someone please give me some general...

I am still struggling with the topic of cardinality, it would be nice if someone could help: 1) http://www.geocities.com/asdfasdf23135/absmath2.jpg In the solutions , they said that{y=ax+b|a,b E R} <-> R2. But I am wondering...what is the actual mapping that gives one-to-one...

1) Consider the xy-plane. Find the cardinality of the set of constructible points on the x-axis. Attempt: Every constructible number is algebraic (i.e. Let A=set of algebraic numbers, C=set of constructible nubmers, then C is a subset of A) and A is countable. => |C|<|A|=|N|...

These are some related questions in my mind, though I am rather confused about them. 1. What does \infty at the "end" of the real number line have to do with \aleph_0, the cardinality of the integers, and C, the cardinality of the continuum? Is \infty equal to one or the other (if such a...

Well, it's a conjecture to me because I don't know (yet) if it's true or false. Let |A|=n, where n is an infinite cardinal. Let B be the collection of all subsets of A with cardinality less than n. Then |B|=n. Is it true first of all? And will the proof be short or long?

According to the Bloch's theorem, the solutions of SE in a periodic potential may be written as superpositions of Bloch waves. But what kind of superpositions are these? There is the continuous wave vector parameter, over which we can integrate just like in forming free wave packets, but what...

I am reading "The linear algebra a beginning graduate student ought to know" by Golan, and I encountered a puzzling statement: Let V be a vector space (not necessarily finitely generated) over a field F. Prove that there exists a bijective function between any two bases of V. Hint: Use...

Prove that the set T of transcendental numbers (numbers that do not satisfy some polynomial equation of positive degree with rational coefficients) has the power of the continuum, i.e. has cardinality c. Here's what I have: Since T is uncountable, then |T|>alephnull . Also, since T is a...

just a cool fact I thought I'd share with anyone who's interested: The set of real values functions on any interval in R has cardinality at least 2^c. Pf: Consider characteristic functions defined on the interval, (a,b). (Note: a characteristic function is a function that can be defined...