Maths WorkshopMaths problems with full solutions

Countability and cardinality

32 problems · Finite and infinite sets, countability of Q, uncountability of R and Cantor's diagonal argument.

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Ex.218Cardinality of finite sets

#finite sets #cardinality #bijection

Countability and cardinality

Ex.219$\mathbb{N}$ is countably infinite

#naturals #countably infinite #identity

Countability and cardinality

Ex.341The integers are countable

#bijection #countability

Countability and cardinality

Ex.416Countable sets

#countable #denumerable #enumeration

Countability and cardinality

Ex.417Hilbert's hotel

#Hilbert hotel #infinite #bijection

Countability and cardinality

Ex.418Infinite sets

#infinite sets #Dedekind #proper subset

Countability and cardinality

Ex.419$\mathbb{Z}$ is countable

#integers #countability #bijection

Countability and cardinality

Ex.420Summary of cardinality

#summary #cardinality

Countability and cardinality

Ex.817The cardinal $\aleph_0$

#aleph zero #notation #cardinality

Countability and cardinality

Ex.818The cardinality of the continuum

#continuum #cardinality #2^aleph0

Countability and cardinality

Ex.819$\mathbb{Q}$ is dense but countable

#rationals #density #countability

Countability and cardinality

Ex.820Cantor's diagonal argument

#diagonal argument #Cantor #uncountability

Countability and cardinality

Ex.821Comparing cardinalities

#comparison #injection #bijection

Countability and cardinality

Ex.822$\mathbb{N}^2$ is countable

#N squared #diagonal #bijection

Countability and cardinality

Ex.823Countable product

#product #countable #product rule

Countability and cardinality

Ex.824$\mathbb{Q}$ is countable

#rationals #countability #surjection

Countability and cardinality

Ex.1463Cantor's theorem

#diagonal argument #sets

Countability and cardinality

Ex.1558Cantor–Bernstein theorem

#Cantor-Bernstein #bijection #injections

Countability and cardinality

Ex.1559The Cantor set is uncountable

#Cantor set #uncountable #ternary

Countability and cardinality

Ex.1560Cantor's theorem

#Cantor theorem #power set #larger cardinality

Countability and cardinality

Ex.1561Finite subsets of $\mathbb{N}$

#finite subsets #countable #naturals

Countability and cardinality

Ex.1562Ordinals vs cardinals

#ordinals #cardinals #distinction

Countability and cardinality

Ex.1563$|\mathcal{P}(\mathbb{N})|=|\mathbb{R}|$

#power set #reals #cardinality

Countability and cardinality

Ex.1564$\mathbb{R}$ is uncountable

#reals #uncountable #Cantor diagonal

Countability and cardinality

Ex.1565Countable union of countable sets

#countable union #diagonalization #countable

Countability and cardinality

Ex.2096Baire category theorem (statement)

#Baire #category #uncountable

Countability and cardinality

Ex.2097Cantor function

#Cantor function #cardinality #devil staircase

Countability and cardinality

Ex.2098Trichotomy of cardinals (statement)

#trichotomy #cardinals #choice

Countability and cardinality

Ex.2099The continuum hypothesis (statement)

#continuum hypothesis #aleph #independence

Countability and cardinality

Ex.2100The generalized continuum hypothesis (statement)

#GCH #generalized continuum #set theory

Countability and cardinality

Ex.2101Infinite subsets of $\mathbb{N}$

#infinite subsets #uncountable #power set

Countability and cardinality

Ex.2102Proof of Cantor–Bernstein–Schröder

#Cantor-Bernstein #proof #bijection

Countability and cardinality