Linear equations, notation, sets, and functions

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.title[
# Linear equations, notation, sets, and functions
]
.author[
### <a href="https://jmclip.github.io/MACSS_math_camp/">MACS 33000</a> <br /> University of Chicago
]

---

# Learning objectives

* Define computational social science
* Connect mathematics and statistics to social scientific inquiry
* Review course logistics
* Explain the pedagogical approach for the camp
* Define basic mathematical notation
* Assess the use of mathematical notation in the rational voter theory
* Define functions and their properties
* Define sets
* Practice root finding
* Define logarithmic and exponential functions
* Practice simplifying power, logarithmic, and exponential functions

---

# Computational social science

* Social science - the scientific study of human society and social relationships

### Disciplines within social science

* Anthropology
* Economics
* History
* Political science
* Psychology
* Sociology

### Interdisciplinary programs

* Comparative human development
* Conceptual and historical studies of science
* Computational social science

---

# Computational social science

* Computational approaches to the social sciences
* Historic definition
* Modern definition

---

# Acquiring CSS skills

* **Computer science**
    * CAPP 30121/122/123 sequence
    * MACS 30500
    
* **Social science**
    * Perspectives sequence (MACS 30000/100/200)
    * Seminars
    * Non-computational courses
    
* **Math/statistics**
    * Mathematical models of social worlds
    * Probability and statistics used across sciences
    * Start with Computational Math Camp
    * Math/stats electives
        * Machine learning
        * Causal inference
        * Bayesian inference
        * Network analysis
        * Deep learning
        * Spatial data science
        * Natural language processing

---

# Math, probability, and statistics

### Mathematics
* Purely abstract
* Based on axioms that are independent from the real world
* Language for expressing structure and relationships

### Probability
* Systematic and rigorous method for treating uncertainty
* "Mathematical models of uncertain reality"
* Derivation of "applied mathematics"

### Statistics
* Collecting and analyzing numerical data in large quantities
* Purpose of inferring proportions in a whole from those in a sample
* Making inferences from data that are not entirely certain

---

# Their uses

* **Mathematical models**
    * Game theory
    * Formal theory
    * Much bigger in economics
    * Defining statistical models and relationships
* **Probability/statistics**
    * Establishing a structure for relationships between variables using data
    * Inferring relationships and assessing their validity

---

# Goals for the math/stats camp

* Survey math and statistical tools that are foundational to CSS
* Review common mathematical notation
* Apply math/statistics methods

**NOTE: we will move FAST -- do your best to keep up and follow along. Come to OH and please check in when/as needed!**

---

# Course logistics

* [Course site](https://github.com/math-camp/course)
* [Canvas course](https://canvas.uchicago.edu/enroll/MB3HTY)
* Gradescope (accessed via Canvas)

## Course staff

* Dr. Jean Clipperton: clipperton@uchicago.edu 
* Saaket Raman: saaket@uchicago.edu
* Xulan Jiang: xulanj@uchicago.edu
* Yanjing Li: yanjingli@uchicago.edu

---

# Prerequisites

* No formal prerequisites
* Prior training in:
    * Linear algebra
    * Calculus
    * Probability theory
    * Statistical inference
* High school/AP/IB training may be sufficient
* We assume prior exposure to the content covered in this camp
* Alternatives to this camp: through PS department (conflicts with computing boot camp)

---

# Evaluation

* Pass/fail (no credit)
* Show up, complete the problem sets satisfactorily (not perfectly), and you'll pass
* The irony of grad school
    * Grades no longer matter (or should not)
    * Learn as much material as possible
    * **If you truly only care about learning material, you'll do well**

---

# Pedagogical approach

* Focus on participation and engagement
* Practice questions integrated
* Regular assignments to check understanding

## Paths to success

* Work together
* Ask for help
* Get quick feedback

---
    
# Why are we doing this?

* This is the magic that makes our models work
* Implicit knowledge: no explicit focus on proofs but understanding assumed

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# Why math is important to social science

* Consistent language to communicate ideas in an orderly and systematic way
* Science uses highly precise language that is not easily interpretable to outsiders
* Mathematics is an effective way to describe our world
* Mathematical notation lets us convey **precision** and minimizes the risk of misinterpretation by other scholars

---

# Example: Rational voter theory

* Rational voters should weigh the rewards vs. costs of voting
    * If it costs more to vote than the rewards gained from the action, then individuals should not vote in elections
    * Difference between cost and reward is defined as the **utility** that the person receives from the act (based on an unknown preference scale)
* How can we codify this model using mathematical notation?

---

# Example: Rational voter theory

`$$R = PB - C$$`

* `$R =$` the utility satisfaction of voting
* `$P =$` the actual probability that the voter will affect the outcome with her particular vote
* `$B =$` the perceived difference in benefits between the two candidates measured in **utiles** (units of utility)
* `$C =$` the actual cost of voting in utiles (e.g. time, effort, money)

--
* What implications does the model provide?
* How would you express them mathematically?

---

# Example: Rational voter theory

Implication | Formal statement
------------|------------------
If individuals do not get enough benefit from voting, they will abstain | The voter will abstain if `$R < 0$`.
Individuals have other things to do on election day (like going to work). If the benefit of voting is not as large as alternative benefits, then individuals will abstain. | The voter may still not vote even if `$R > 0$` if there exist other competing activities that produce a higher `$R$`.
Most elections have many, many ballots cast. There is no point to voting since any individual ballot is unlikely to change the outcome of the election. Thus, everyone should abstain. | If `$P$` is `$\approx$` 0, then `$PB<C$`, so `$R<0$` and the voter abstains.

---

# Example: Rational voter theory

* Paradox of voting
* Simple model, but incredibly impactful
* Pure mathematical model may not be fully accurate, but allows us to delve deeper into the paradox
    * How to measure `$B$`
    * How to measure `$C$`

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# **Questions?**

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# Sets

* Collection of objects

$$
`\begin{aligned}
A & = \{1, 2, 3\} \nonumber  \\
B  & = \{4, 5, 6\}\nonumber  \\
C  & = \{ \text{First year cohort} \} \\
D & = \{ \text{U of Chicago SIPs} \}
\end{aligned}`
$$

---

# Sets

* If `$A$` is a set, we say that `$x$` is an element of `$A$` by writing `$x \in A$`
* If `$x$` is not an element of `$A$` then, we write `$x \notin A$`

* `$1 \in \{ 1, 2, 3\}$`

* `$4 \in \{4, 5, 6\}$`

* `$\text{Xulan} \notin \{ \text{First year cohort} \}$`

* `$\text{Jean} \in \{ \text{U of Chicago SIPs} \}$`

Why do we care about sets?

* Sets are necessary for probability theory
* Defining **set** is equivalent to choosing population of interest (usually)

---

# Subsets

* If `$A$` and `$B$` are sets, then we say that `$A = B$` if, for all `$x \in A$` then `$x \in B$` and for all `$y \in B$` then `$y \in A$`
* Test to determine equality:
    * Take all elements of `$A$`, see if in `$B$`
    * Take all elements of `$B$`, see if in `$A$`

-----------

* If `$A$` and `$B$` are sets, then we say that `$A \subseteq B$` if, for all `$x \in A$`, then `$x \in B$`
    * What is the difference between the definitions?

---

# Set builder notation

Some famous sets:

* `$\mathbb{N} = \{1, 2, 3, \ldots \}$`
* `$\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots \}$`
* `$\mathbb{R} = \mbox{Real numbers}$`

Use **set builder notation** to identify subsets:

* `$[a, b]  = \{x: x \in \mathbb{R} \text{ and } a \leq x \leq b \}$`
* `$(a, b]  = \{x: x \in \mathbb{R} \text{ and } a < x \leq b \}$`
* `$[a, b)  = \{x: x \in \mathbb{R} \text{ and } a \leq x <  b \}$`
* `$(a, b)  = \{x: x \in \mathbb{R} \text{ and } a < x < b \}$`
* `$\emptyset$`

---

# Union

* `$A$` and `$B$` are sets
* New set that contains all elements in set `$A$` *or* in set `$B$`

`$$\begin{aligned}C & = A \cup B  \\   & = \{x: x \in A \text{ or } x \in B \}\end{aligned}$$`

* `$A = \{1, 2, 3\}, B = \{3, 4, 5\}$`, then `$C = A \cup B  = \{ 1, 2, 3, 4, 5\}$`
* `$D  = \{\text{First year cohort} \}, E = \{\text{Me} \}$`, then `$F = D \cup E = \{ \text{First year cohort, Me} \}$`

Think of the union as a list of the elements that are present (not an accounting of their number).

---

# Intersection

* New set that contains all elements in set `$A$` *and* set `$B$`

`$$\begin{aligned}C  & = A \cap B \\  & = \{x: x \in A \text{ and } x \in B \}\end{aligned}$$`

* `$A =\{1, 2, 3\}, B = \{3, 4, 5\}$`, then, `$C = A \cap B = \{3\}$`
* `$D = \{\text{First year cohort} \}, E = \{\text{Me} \}$`, then `$F = D \cap E = \emptyset$`

---

# Some facts about sets

1. `$A \cap B = B \cap A$`
1. `$A \cup B = B \cup A$`
1. `$(A \cap B) \cap C = A \cap (B \cap C)$`
1. `$(A \cup B) \cup C = A \cup (B \cup C)$`
1. `$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$`
1. `$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$`

---

# Ordered pairs

`$$(a, b)$$`

* Two sets `$A$` and `$B$`
* Cartesian product

`$$A \times B = \{(a, b): a \in A \text{ and } b \in B \}$$`

--
  
Example: `$A = \{1, 2\}$` and `$B = \{3, 4\}$`, then,

`$$A \times B = \{ (1, 3), (1, 4), (2, 3), (2, 4) \}$$`

---

# Relation

* Relation
* Function `$F$` - a relation such that,

$$
    (x, y) \in F ; (x, z) \in F \Rightarrow y = z
    $$

* `$F(x)$`
    * `$x \in \mbox{Domain} \, F$`
    * `$F(x) \in \mbox{Codomain} \, F$`
    * `$F:A \rightarrow B$`

---

# Functions

---

# Functions

---

# Functions

---

# Relation vs. function

* Mathematical function - a mapping which gives a correspondence from one measure onto exactly one other for that value
* Mapping from one defined space to another, such as `$F \colon \mathbb{R} \rightarrow \mathbb{R}$`

`$$F(x) = x^2 - 1$$`
    
    * Maps `$x$` to `$F(x)$` by squaring `$x$` and subtracting 1

## Not a function

* All functions are relations, but not all relations are functions

---

# Not a function

---

# Not a function

---

# Two properties of functions

`$$F(x) = y$$`

1. A function is **continuous** if it has no gaps in its mapping from `$x$` to `$y$`
1. A function is **invertible** if its reverse operation exists:

`$$F^{-1}(y) = x, \text{where } F^{-1}(F(x)) = x$$`

---

# Non-continuous

`$$F(x) = \left\{
        \begin{array}{ll}
            \frac{1}{x} & \quad x \neq 0 \\
            0 & \quad \text{otherwise}
        \end{array}
\right.$$`

---

# Non-invertible

---

# Two properties of functions

* We want functions to be continuous **and** invertible
* Non-continuous functions
* Non-invertible functions
    * Can make invertible by restricting the **domain**

---

# Restrict the domain

---
# Piecewise functions

* Restricting the domain can produce a *piecewise* function: one function built from two or more different rules or equations with non-overlapping domains

### Example: `$|x|$` is piecewise (on either side of zero)

---
# Function Composition

You can also compose or decompose functions by separating them:

`$$f(g(x)) = (f \circ g)(x)$$`

* Work from the inside out: first compute `$g(x)$`, then feed the result into `$f$`

Example: `$f(x) = x + 1$` and `$g(x) = x^2$`

`$$f(g(2)) = f(4) = 5$$`

`$$g(f(2)) = g(3) = 9$$`

* **Order matters:** `$f(g(x)) \neq g(f(x))$` *usually*

---

# Quadratic functions

`$$y = ax^2$$`

---

# Quadratic functions

`$$y = ax^2 + bx + c$$`

## Quadratic equation

* `$f(x) = 0$` where `$f$` is a quadratic function

--
* To **solve the equation**, we need to determine the value of `$x$` which satisfies the equation

---

# Quadratic equation

.pull-left[

$$
`\begin{aligned}
x^2 - 7 &= 0 \\
x^2 &= 7 \\
x &= \pm \sqrt{7}
\end{aligned}`
$$

]

.pull-right[

]

---

# Quadratic equation

.pull-left[

$$
`\begin{aligned}
-3x^2 + 30x - 27 &= 0 \\
-3 (x^2 - 10x + 9) &= 0 \\
-3(x - 9)(x - 1) &= 0 \\
(x - 9)(x - 1) &= 0 \\
x &= 1, 9
\end{aligned}`
$$

]

.pull-right[

]

---

# Quadratic formula

`$$ax^2 + bx + c = 0$$`
  
* Quadratic formula

`$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$`

---

# Quadratic formula

`$$x^2 + x - 12 = 0$$`

$$
`\begin{aligned}
x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\
&= \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times -12}}{2 \times 1} \\
&= \frac{-1 \pm \sqrt{1 - (-48)}}{2} \\
&= \frac{-1 \pm \sqrt{49}}{2} \\
&= \frac{-1 \pm 7}{2} \\
&= 3, -4 \\
\end{aligned}`
$$

---
# Quadratic formula, soln

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# Logarithms and exponential functions

* Important component to many mathematical and statistical methods in social science
* Exponents
* Logarithms

---

# Functions with exponents

`$$f(x) = x \times x = x^2$$`

`$$f(x) = x \times x \times x = x^3$$`

---

# Common rules of exponents

* `$x^0 = 1$`
* `$x^1 = x$`
* `$\left ( \frac{x}{y} \right )^a = \left ( \frac{x^a}{y^a}\right ) = x^a y^{-a}$`
* `$(x^a)^b = x^{ab}$`
* `$(xy)^a = x^a y^a$`
* `$x^a \times x^b = x^{a+b}$`

---

# Logarithms

* Class of functions
* `$\log_{b}(x) = a \Leftrightarrow b^a = x$`
* What number `$a$` solves `$b^a = x$`

---

# Base 10

`$$\log_{10}(100) = 2 \Rightarrow 10^2 = 100$$`

`$$\log_{10}(0.1) = -1 \Rightarrow 10^{-1} = 0.1$$`

---

# Base 2

`$$\log_{2}(8) = 3 \Rightarrow 2^3 = 8$$`

`$$\log_{2}(1) = 0 \Rightarrow 2^0 = 1$$`

---

# Base `$e$`

* Natural logarithm

`$$\log_{e}(e) = 1 \Rightarrow e^1 = e$$`

* Natural logarithms are incredibly useful in math
* Often `$\log()$` is assumed to be a natural log (good to assume for homework, too!)
* Also seen as `$\ln()$`

---

# Rules of logarithms

* `$\log_b(1) = 0$`
* `$\log(x \times y) = \log(x) + \log(y)$`
* `$\log(\frac{x}{y}) = \log(x) - \log(y)$`
* `$\log(x^y) = y \log(x)$`

---
class: center, middle, inverse
# Pause + Gradescope

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# Gradescope
Online system for grading.

Steps:

* Download assignment (pdf or tex file)
* Complete assignment by deadline (due that night)
* Upload to Gradescope (navigate through Canvas)
* Label questions
* Yes, it is annoying and I am sorry!

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# Gradescope: labeling

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# Gradescope: labeling in action
How to label:

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# Gradescope: views

Please label or we get stuck with this!!

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# Reminders

* Assignment due TONIGHT (grace period -- look at canvas)
* Canvas FYI: due date vs grace period
* LABEL THINGS