Cardinality Definition and 174 Threads

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...

I'm fairly sure that the intervals (0,1) and [0,1] of real numbers have the same cardinality, but I can't think of a bijection between them. Any thoughts?

Hello, I have an aficionado curiosity, so please bear with me. As you know, bags are sets where repeated elements are allowed. Imagine the following funny representation for a bag: instead of repeating elements, we use a set of ordered pairs, containing each distinct item plus a count of how...

Homework Statement Let X be a finite set with n elements. Prove that P(X) has 2^n elements. <This is an extra credit problem for a summer class I'm taking.> Homework Equations P(X) is the power set of X, the set of all possible subsets of X. The principle of induction. The...

How can I prove that \left| {\left\{ {A \subset \mathbb{N}:\left| A \right| \in \mathbb{N}} \right\}} \right| = \left| \mathbb{N} \right| ?

[Resolved][Sets] Cardinality problem Homework Statement let A be a Set of all natural numbers from 1 to 6000 that are divsible by 3 or 7 but not 105. 1.What is the cardinality of A? 2.How many numbers in A give 2 as the remained of division by 3. Homework Equations The Attempt at...

i need to show that there exists a class of sets A which is a subset of P(Q) such that it satisfies: 1) |A|=c (c is the cardinality of the reals) 2) for every A1,A2 which are different their intersection is finite (or empty). basically i think that i need to use something else iv'e proven...

i need to find the cardinality of set of continuous functions f:R->R. well i know that this cardinality is samaller or equal than 2^c, where c is the continuum cardinal. but to show that it's bigger or equals i find a bit nontrivial. i mean if R^R is the set of all functions f:R->R, i need to...

i need to find the cardinality of the set of all concave polygons. i know that each n-polygon is characterized by its n sides, and n angles, but i didn't find its cardinality, for example we can divide this set to disjoint sets of: triangles,quandrangulars, etc. we can characterize the...

Homework Statement Prove that the union of c sets of cardinality c has cardinality c. Homework Equations The Attempt at a Solution Well, I could look for a one-to-one and onto function... maybe mapping the union of c intervaks to the reals, or something? I know how to demonstrate...

Suppose X is a set consisting of squares with the property that any addition with elements of X (where no two are the same) gives a square (might not be in X). How many elements can X have?

Find the cardinality and dimension of the vector space \mathbb{Z}^{3}_{7} over \mathbb{Z}_{7}. \mathbb{Z}^{3}_{7} = \{ (a,b,c) \; | \; a,b,c \in \mathbb{Z}_{7} \}. Then since \mathbb{Z}_{7} is a field 1 \cdot a = a \; \forall \; a, so B = \{ (1,0,0), (0,1,0) , (0,0,1) \} is a basis of...

Just come across this question on a problem sheet and it's got me rather confused! You have to prove that |[0,1]|=|[0,1)|=|(0,1)| without using Schroeder-Bernstein and using the Hilbert Hotel approach. After looking at the Hilbert Hotel idea I can't really understand how this helps! This...

So the problem, and my partial solution are in the attached PDF. I would like feedback on my proof of the first statement, if it is technically correct and if it is good. Any ideas as to how I can use/generalize/extend the present proof to proof the second statement, namely that E (the Cantor...

Kind of trivial result, but thought it might be interesting. This is part of a wider development which will be described further, either here or in another thread. Statement: "A Point in Spacetime has the Cardinality of the Continuum" Justification: Time can play a really neat...

Show that they are the same number of points in a line, in a plane and in the space. I have one more question: Which set has a cardinal number greater than the continuum. Why? Thanks in advance.

O.K this has been bugging me all night since I first thought of it. How would I show the sets, \left\{ 0 < x < 1 \left| x \in \mathbb{R}\left\} \left\{ 0 < x \leq 1 \left| x \in \mathbb{R}\left\} Have equal cardinality?

Prove that the set of complex numbers has the same cardinality as the reals. What I did was say that a + bi can be written as (a, b) where a, b belong to real. Which essentially means i have to create a bijection between (a, b) and z (where z belongs to real). Suppose: a = 0.a1a2a3a4a5...