Functions Sequence And Summations

Functions-Sequence-and-Summations

Functions

  • A function $f$ from $A$ to $B$ is a subset of $A×B$ which satisfies the following two conditions
      1. $\forall x (x \in A \rightarrow \exists y (y \in B \wedge (x,y) \in f))$^[For all elements in A]
      1. $(((x_1,y_1) \in f \wedge (x_1,y_2)\in f) \rightarrow y_1 =y_2)$ ^[One and only one y in B for each x in A]

One-to-One Functions (Injections)

  • $f$ is one-to-one iff:
    • For $a,b \in A$, if $a\not=b$ then $f(a) \not= f(b)$
  • One-to-One function is also called injective function ^[(单射)]

Onto Functions (Surjections)

  • $f$ is onto iff:
    • $\forall y \in B (\exists x (x \in A \wedge f(x) = y))$
  • Onto function is also called surjective function ^[(满射函数) 值域y是满的,每个y都有x对应,不存在某个y没有x对应的情况]

One-to-One and Onto Functions (Bijections)

  • $f$ is a bijective function iff
    • $f$ is both onto and one-to-one
  • One-to-One and onto function is also called bijective function ^[(双射函数)]

Relationship between domain, co-domain and range

  • Injection: $ A = f(A) <= B $
  • Surjection: $ f(A) = B $, But $ A $ may not equal to $ f(A) $
  • Bijection: $ A = f(A) = B $

Image, Pre-image and Range

  • 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 $f$.

Inverse Function

  • $f:$ $A \rightarrow B$ is a bijection.
    • The inverse of $f$ is bijection $f^{-1}:$ $B \rightarrow A $ such that $f^{-1}(f(x))=x$ for all x $\in A$
    • if $f(x) = y$ then $f^{-1}(y)=x$

Composition Function

If $f$ is a function from $A$ to $B$ and $g$ is a function from $B$ to $C$, then $g * f(x) (g(f(x)))$ is the composition of g and f

Two special functions

  • Floor Function
    • Denoted $\lfloor x\rfloor $ eg. $\lfloor 2.9\rfloor =3$
  • Ceiling Funcation
  • Denoted $\lceil x\rceil$ eg. $\lceil -2.9 \rceil=-2$

Sequences

  • 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 $S$, denoted ${a_n}$. The integers determine the positions of the elements in the list.

Summations

  • Sequences are very useful in summations.
  • A summation is the value of the sum of the terms of a sequence. \(a_1+a_2+a_3+...+a_n=\sum^{n}_{j=1}a_j=\sum_{1\leq j \leq n}a_j=\sum^n_{j=1}a_j=\sum^n_{k=1}a_k\)

Some Special Summations

  • A geometric series is a summation of a geometric progression
    • Geometric progression: $a,ar,ar^2,…$
    • Geometric series: $\sum^n_{j=0}ar^j$
  • A harmonic series is the summation of a harmonic progression
    • Harmonic progression: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},…,\frac{1}{n}$
    • Harmonic series:$\sum^n_{j=1}\frac{1}{j}$

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