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

• $\mathbb{N}$ : the set of all natural numbers
• $\mathbb{Z}$ : the set of integers
• ${\mathbb{Z}}^{+}$ : the set of all the positive integers
• $\mathbb{Q}$: the set of all rational numbers
• $ℝ$: the set of all the real numbers
• $ℂ$: 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

• $A\in B$ : A is in or is an element of B
• $A\notin B$ : 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

$P(S)=A|A\subseteq S$

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 $S_1,S_2,\cdots ,S_n(denoted{S}_1\times S_2\times \cdots \times S_n)$
$S_1\times S_2\times \cdots \times S_n=\{(a1,a2,\dots ,an)|a_1\in S_1\wedge a_2\in S_2\wedge \cdots \wedge a_n\in S_n\}$

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 $y=f(x)$ from set A to set B, then

• 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, $f(A)=f(x)\parallel x\in A$

injective function（单射）

f is one-to-one

urjective function (满射)

Onto function :$\forall y\in B(\exists x(x\in A\wedge f(x)=y))$

bijective function (双射)

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

Floor functions

• Denoted $\lfloor x\rfloor$
• The largest integer less than or equivalent to x

Ceiling functions

• Denoted $\lceil x\rceil$
• 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 {$a_n$}. The integers determine the positions of the elements in the list

Summations 求和

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

Special Summations

Geometric series 等比数列和

$\sum _{j=0}^{n}a{r}^j$

harmonic series

$\sum _{j=1}^n\frac{1}{j}$