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Cardinality and infinity

Countable and uncountable sets, Cantor's diagonal argument, the continuum.

Two sets have the same cardinality when a bijection links them. The naturals, integers and rationals are all countable; the reals are not — write any list of them and the diagonal argument builds a real not on the list. Picture it: the zig-zag through the grid of fractions that lists every rational. Think it: the continuum hypothesis — is there a size between ℕ and ℝ? — is independent of the standard axioms.

Töötatud näide: sum of 1/2^n for n = 0 to oo

Calculus · Series-And-Sums

Sum 2^(-n) for n = 0 to oo

\sum_{n=0}^{\infty} 2^{- n}

Samm-sammult

  1. \sum_{n=0}^{\infty} 2^{- n}

    Write the sum out.

  2. 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots

    The first few terms.

  3. S_{3} \approx 1.7500, S_{6} \approx 1.9688, S_{11} \approx 1.9990, S_{51} \approx 2.0000

    Partial sums approach the limit.

  4. = 2

    Infinite series: this converges, and the closed form is the limit of the partial sums.

Vastuse avalikustamine
2

Symbols used here

\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\approx
approximately equal
Equal to the precision shown, not exactly.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
\neq
not equal
The two sides are different.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.

How to: Cardinality and infinity

  1. Write the sum out.
  2. The first few terms.
  3. Partial sums approach the limit.
  4. Infinite series: this converges, and the closed form is the limit of the partial sums.

Questions people ask

Are some infinities bigger than others?

Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.

What is the difference between a relation and a function?

A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.

Proovi ise.

Rohkem Set Theory & Logic

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