Analysis - 解析学(1)

This article collects basic definitions used in set theory, algebra, and analysis.

1. Set

A set is a well-defined collection of distinct objects, called its elements. A statement such as $x \in X$ asserts that $x$ is an element of the set $X$.

2. Extensional and intensional definitions

A set can be described by listing its elements (extensional definition) or by stating the property that its elements satisfy (intensional definition). For example, the set of lowercase English letters is

\[\{a,b,c,\ldots,z\}.\]

3. Subset

A set $X$ is a subset of $Y$, written $X \subseteq Y$, when every element of $X$ is also an element of $Y$:

\[X \subseteq Y \iff \forall x\,(x \in X \Rightarrow x \in Y).\]

4. Union

The union of $X$ and $Y$ contains the elements that belong to at least one of the sets:

\[X \cup Y = \{x \mid x \in X \text{ or } x \in Y\}.\]

5. Intersection

The intersection of $X$ and $Y$ contains the elements common to both sets:

\[X \cap Y = \{x \mid x \in X \text{ and } x \in Y\}.\]

6. Set difference

The difference of $X$ and $Y$ is the set of elements in $X$ that are not in $Y$:

\[X \setminus Y = \{x \in X \mid x \notin Y\}.\]

7. Complement

Relative to a fixed universe $U$, the complement of $X \subseteq U$ is

\[X^c = U \setminus X.\]

8. Empty set

The empty set, written $\emptyset$, has no elements. It is a subset of every set.

9. Power set

The power set of $X$ is the set of all subsets of $X$:

\[\mathcal{P}(X) = \{A \mid A \subseteq X\}.\]

10. Cartesian product

The Cartesian product of $A$ and $B$ is the set of ordered pairs

\[A \times B = \{(a,b) \mid a \in A,\ b \in B\}.\]

For sets $X_1,\ldots,X_n$, their product is

\[X_1 \times \cdots \times X_n = \{(x_1,\ldots,x_n) \mid x_i \in X_i\text{ for every }i\}.\]

When every $X_i$ is the same set $X$, this product is written $X^n$.

11. Symmetric difference

The symmetric difference contains elements that belong to exactly one of $A$ and $B$:

\[A \triangle B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B).\]

12. Groups and order

A group is a nonempty set $G$ with a binary operation $\cdot:G\times G\to G$ satisfying associativity, having an identity element, and giving every element an inverse. Explicitly, for all $x,y,z\in G$:

• $(x\cdot y)\cdot z = x\cdot(y\cdot z)$;
• there is an identity $e\in G$ with $e\cdot x=x\cdot e=x$;
• for every $x\in G$, there is $x^{-1}\in G$ with $x\cdot x^{-1}=x^{-1}\cdot x=e$.

A semigroup has an associative binary operation. A monoid is a semigroup with an identity element. A group is abelian if its operation is commutative. The order of a finite group is its number of elements.

13. Algebraic and transcendental numbers

A complex number is algebraic if it is a root of a nonzero polynomial with rational coefficients. A complex number that is not algebraic is transcendental.

14. Function

A function $f:X\to Y$ assigns exactly one element of $Y$ to each element of $X$. The set $X$ is the domain, and $Y$ is the codomain. The image (or range) is

\[f(X)=\{f(x)\mid x\in X\}\subseteq Y.\]

The notation $x\mapsto f(x)$ describes the assignment.

15. Injective, surjective, and bijective functions

A function $f:X\to Y$ is:

• injective if $f(x_1)=f(x_2)$ implies $x_1=x_2$;
• surjective if for every $y\in Y$ there is an $x\in X$ such that $f(x)=y$;
• bijective if it is both injective and surjective.

16. Inverse function

A function $f:X\to Y$ has an inverse function $f^{-1}:Y\to X$ exactly when it is bijective. The inverse satisfies $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$.

17. Equinumerous sets

Sets $A$ and $B$ are equinumerous if there is a bijection from $A$ to $B$. This is written $A\sim B$.

18. Cardinality and countability

19. Comparing cardinalities