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Math and science::Analysis::Tao::03: Set theory

# Cardinality of sets

#### Equal cardinality

We say that two sets $$X$$ and $$Y$$ have equal cardinality iff there exists a bijection $$f : X \rightarrow Y$$ from $$X$$ to $$Y$$.

#### Cardinality n

Let $$n$$ be a natural number. A set $$X$$ is said to have cardinality n if it has equal cardinality with the set $$\{i \in \mathbb{N} : 1 \ge i \ge n \}$$. We also say that such a set has $$n$$ elements.

#### Finite sets

A set is finite iff it has cardinality $$n$$ for some natural number $$n$$; otherwise, the set is called infinite.

Notation: if $$X$$ is a finite set, we use $$\#(X)$$ to denote the cardinality of $$X$$.

### Example

The set of natural numbers is infinite.

Sets with equal cardinality can have one contain the other. For example, there is a bijection between the set of natural numbers and the set of even natural numbers, $$f: X \rightarrow Y$$, $$f(n) := 2n$$.

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