<commit_before>name: inverse
layout: true
class: center, middle, inverse
---
# STAT 430: Lecture 1 
## Why Computer Scientists Study Statistics
### Chapter 1: Introduction
.footnote[Course page: [imouzon.github.io/stat430](https://imouzon.github.io/stat430)]
---
# A Little Background
## A Quick Review of Sets, Functions, and Chaos
---
layout: false
.left-column[
  ## Sets
  ### Terms and Notation
]
.right-column[
## Terminology and Notation for Sets

A _set_ is finite or infinite collection of objects in which order has no significance and multiplicity is also generally ignored.

We generally use capital letters, in bold face when typed, for sets such as 
\\[
\mathbf{A}, \mathbf{B}, \mathbf{A\_1}, \mathbf{A\_2}, \ldots, \mathbf{A\_n}
\\] 
and so on.
When a set is defined explicitly, we will generally use the following notation
\\[
\mathbf{A} = \\{a\_1, a\_2, \ldots, a\_n\\}
\\]
\\[
\mathbf{B} = \\{b\_1, b\_2, \ldots \\}
\\]
Or describe a rule that all elements of the set must fit
\\[
\mathbf{A} = \\{x > 0 : x\mod 3 = 5\\}
\\]
The objects collected in the set are referred to as _elements_ of the set.
We can indicate an object is an element of set by writing
\\(
a \in \mathbf{A}
\\)
]
---
layout: false
.left-column[
  ## Sets
  ### Terms and Notation
  ### Special cases
]
.right-column[
## Special Sets
Certain sets are dealt with so regularly that they have their own standard representations.

-  \\(\mathbf{S}\\) is the *universe*, the set containing all the elements being considered in a situation
-  \\(\emptyset\\) is the *empty set* or the set containing no elements.
-  the real numbers: \\(\mathbb{R} = (-\infty, \infty)\\)
-  the integers: \\(\mathbb{Z} = \\{\ldots, -2, -1, 0, 1, 2, \ldots\\}\\)
-  the positive real numbers: \\(\mathbb{R^{+}} = (0, \infty)\\)
-  the positive real numbers include 0: \\(\mathbb{R^{+}\_0} = [0, \infty)\\) <!----]-->

## Special Circumstances
Because the concept of sets has such wide application, many fields have special notation that identifies a specific set as having special properties or roles:

-  \\(\mathcal{F}\\) is used in measure theory to denote the set of "measureably sets"
-  \\(\mathcal{B}_r(\mathbf{p})\\) is used in topology as the ball of radius \\(r\\) centered at \\(\mathbf{p}\\).

]
---
layout: false
.left-column[
  ## Sets
  ### Terms and Notation
  ### Special cases
  ### Operations
]
.right-column[
## Set Operations

**Complimention**

_Motivation: I want to say "everything that is not in the set"_

The **compliment** of a set \\(\mathbf{A}\\) is a set containing all the elements that are not elements of \\(\mathbf{A}\\).

\\[
\mathbf{A}^c = \\{x : x \notin \mathbf{A}\\}
\\]

_examples_: 

If \\(\mathbf{S} = \\{a, b, c, d, e\\}\\) and \\(\mathbf{A} = \\{a, b, c\\}\\) then \\(\mathbf{A}^c = \\{d, e\\}\\)

_note_: \\(\mathbf{S}^c = \emptyset\\)
]<commit_msg>format fixes to lecture 1<commit_after>name: inverse
layout: true
class: center, middle, inverse
---
# STAT 430: Lecture 1 
## Why Computer Scientists Study Statistics
### Chapter 1: Introduction
.footnote[Course page: [imouzon.github.io/stat430](https://imouzon.github.io/stat430)]
---
# A Little Background
## A Quick Review of Sets, Functions, and Chaos
---
layout: false
.left-column[
  ## Sets
  ### Terms and Notation
]
.right-column[
## Terminology and Notation for Sets

A _set_ is finite or infinite collection of objects in which order has no significance and multiplicity is also generally ignored.

We generally use capital letters, in bold face when typed, for sets such as 
\\[
\mathbf{A}, \mathbf{B}, \mathbf{A\_1}, \mathbf{A\_2}, \ldots, \mathbf{A\_n}
\\] 
and so on.
When a set is defined explicitly, we will generally use the following notation
\\[
\mathbf{A} = \\{a\_1, a\_2, \ldots, a\_n\\}
\\]
\\[
\mathbf{B} = \\{b\_1, b\_2, \ldots \\}
\\]
Or describe a rule that all elements of the set must fit
\\[
\mathbf{A} = \\{x > 0 : x\mod 3 = 5\\}
\\]
The objects collected in the set are referred to as _elements_ of the set.
We can indicate an object is an element of set by writing
\\(
a \in \mathbf{A}
\\)
]
---
layout: false
.left-column[
  ## Sets
  ### Terms and Notation
  ### Special cases
]
.right-column[
## Special Sets
Certain sets are dealt with so regularly that they have their own standard representations.

-  \\(\mathbf{S}\\) is the *universe*, the set containing all the elements being considered in a situation
-  \\(\emptyset\\) is the *empty set* or the set containing no elements.
-  the real numbers: \\(\mathbb{R} = (-\infty, \infty)\\)
-  the integers: \\(\mathbb{Z} = \\{\ldots, -2, -1, 0, 1, 2, \ldots\\}\\)
-  the positive real numbers: \\(\mathbb{R^{+}} = (0, \infty)\\)
-  the positive real numbers include 0: \\(\mathbb{R^{+}\_0} = [0, \infty)\\) <!----]-->

## Special Circumstances
Because the concept of sets has such wide application, many fields have special notation that identifies a specific set as having special properties or roles:

-  \\(\mathcal{F}\\) is used in measure theory to denote the set of "measureably sets"
-  \\(\mathcal{B}_r(\mathbf{p})\\) is used in topology as the ball of radius \\(r\\) centered at \\(\mathbf{p}\\).

]
---
layout: false
.left-column[
  ## Sets
  ### Terms and Notation
  ### Special cases
  ### Operations
]
.right-column[
## Set Operations

**Complimentation**

_Motivation: I want to say "everything that is not in the set"_

The **compliment** of a set \\(\mathbf{A}\\) is a set containing all the elements that are not elements of \\(\mathbf{A}\\).

\\[
\mathbf{A}^c = \\{x : x \notin \mathbf{A}\\}
\\]

_examples_: 

If \\(\mathbf{S} = \\{a, b, c, d, e\\}\\) and \\(\mathbf{A} = \\{a, b, c\\}\\) then \\(\mathbf{A}^c = \\{d, e\\}\\)

_note_: \\(\mathbf{S}^c = \emptyset\\)
]
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