Cardinality Definition and 167 Threads

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...