# SFSS

## Sets

- A set is a collection of objects
- Sets are used to group objects together

### Three ways to express the members in a set

```
- List all the members
- Use predicates
- Use suspension(省略号)points(must be inferred)
```

### universal set

: the set of all natural numbers : the set of integers : the set of all the positive integers : the set of all rational numbers : the set of all the real numbers : the set of all complex numbers

### Venn Diagrams

- two basic shapes
- A rectangle: indicates the universal set
- Circles or other shapes: indicate normal sets

### Elements and Sets

: A is in or is an element of B : A is not in or is not an element of B

### Subsets

- Subsets
- Proper subsets(真子集)
- Empty sets

### Cardinality

number of distinct elements in a set

The cardinality of a set s is denoted as |s|

### Power Sets

#### Theorem of Power Sets:

$ if |S| = n, then |P(S)| = 2^n$

### Ordered n-tuple

- The form (1, 2, … , ) or < 1, 2, … , >
- (1,2) not equal to (2,1)

### Cartesian Product(笛卡尔乘积)

Cartesian product of

### Disjoint Sets

- If A ∩ B = ∅ then A and B are disjoint.
- If A ∩ B ≠ ∅ then A and B are overlapped.

## function

### conditions

A function from to is a subset of × which satisfies the following two conditions

1.$ ∀ x(x ∈ A → ∃ y(y ∈ B ∧ (x,y) ∈f)) $

2. $ (((x_1,y_1 ) ∈ f ∧ (x_1,y_2 ) ∈ f) → y_1 = y_2)$

### Image, Pre-image and Range(值域)

If

- y is called the image of x under f
- x is called a pre-image of y
- the set of all the images of the elements in the domain under is called the range of f,

### injective function（单射）

f is one-to-one

### urjective function (满射)

Onto function :

### bijective function (双射)

[One-to-One and onto function] is also called bijective function

### Floor functions

- Denoted
- The largest integer less than or equivalent to x

### Ceiling functions

- Denoted
- The smallest integer greater than or equivalent to x

## Sequences 数列

Sequences are ordered lists of elements. A sequence is a function from a subset of the set of integers ({0, 1, 2, 3, … } or {1, 2, 3, … }) to a set , denoted {

## Summations 求和

A summation is the value of the sum of the terms of a sequence.