Unit 14 Basic Structures

What is a basic structure

Sets

• Stets are one of the building blocks for counting

• A set is an unordeerd collection of objects

  • Objects in the set are called elements (or members) and they are said to contain its elements

• Subsets

  • To show if sub set we use $\subset$ like $A\subset B$ and A is part of B

  • Proper subset

    • if something is a proper subset it must be smaller than the parent subset

• Carnality

  • it just tells you how many distinct elements of A

    • $|(1,2,3)|=3$ (3 distinct elements)

• The notations is $a\in A$ denotes that

• Describing a set

  • Roster method

    • $S=(a,b,c)=(c,b,b,c,a)$
      • they are equal as long as the have the same elements!

    • Ex.f

      • set of all vowels in the english alphabet:
        • $V=(a,e,i,o,u)$

• tuples

  • They are ordered sets

Functions

• Definition

  • A and B are nonempty sets, then a functions that assigns A to B, denoted $f:A\to B$ is an assignment

    • you can have mutiple input to the same output (look at x^2)

• Injections

  • A function that is one to one cant be $f(a)=f(b),\text{ implies }a=b$ in the domain of A

• surjection

• Bijection

• Composition

  • basicly a function within a function

Sequence and summations

• sequences

  • ordered list of numbers

  • A function from a subset of the integers

  • the notation $a_n$ is

  • this is just like Series

• Geometric series

  • sums of terms of geometric progressions

    • \large \sum_{j=0}^{n}ar^{j}=(\begin{align}\frac{{ar^{n+1}-a}}{r-1},r=/=1\\\frac{n(n+1)}{2},r=1\\\end{align}

• Product notation

  • Big ass pi