一阶谓词逻辑

命题

非真即假，陈述句，唯一的真值。

如果满足上述条件而暂时无法确定真值也算，明天是晴天、存在外星人都是命题。

Overview

Given the equivalence relation on A set , we can divide into several equivalence classes by . These equivalence classes form a partition of a set.

Conversely, given a partition of a set, we can define an equivalence relation: if two elements belong to the same subset, then they are equivalent. This gives us an equivalence relation.

Theorem

Let be a group, any subgroup of , then the number of left (right) cosets of in is called the index of in , denoted as .

Lagrange's Theorem：

is a finite group, any subgroup of , then .

Proof: Let , then there are distinct left cosets of , denote , then is a partition of (Partition is not covering, cosets in are not overlapped), at this time we have

Definition

Let be a subgroup of (i.e., ). If for all , , then is called a normal subgroup of (or an invariant subgroup), denoted as .

In this case, the left and right cosets are simply called cosets.

The two trivial subgroups of , and itself, are normal subgroups.

Overview

Review the definition of symmetry group:

Let be an arbitrary set. . The group formed by the composition of such permutations is called the symmetric group on . -

Why is it called a symmetric group? This is because such groups describe the symmetry of objects. - In nature and real life, many objects have symmetry, such as isosceles triangles, squares, and regular polyhedra in geometric shapes. The symmetry of these shapes can be described as a transformation in the plane or space where the image of the shape coincides with itself.

Overview

The left and right cosets of a normal subgroup are the same, and the set of all cosets forms a partition of the group. Using the same equivalence relation to form the quotient group, it is denoted by . A group can be defined on , called the quotient group.

is a normal subgroup of . Let the set which is the set formed by all cosets of in .

Overview

Group theory: have only one binary operation, can't describe relationships between multiple operations.

Ring and field: have two binary operation, they're based on group theory.

Overview

Correspondence with group theory：

Subgroup - Subring

normal subgroup - ideal

quotient group - quotient ring

definition

In group , if there's an element , makes for , exists satisfies:

then is a Cyclical group, denoted as , and a is a generator of . The set of all generators of is called the generator set of ​.

That means, every element of can be represented by an integer power of ​.

definition

Let and are groups, mapping, and

, have (Homomorphic equation),

then is group homomorphic mapping from to ​.

When ​ is injective, surjective, and bijective, respectively, it is called a single group homomorphism, a full group homomorphism, and a group isomorphism, respectively.