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Functions - Mapping from Sets to Sets

Course video 16 of 34

This module builds vocabulary for graphing functions in the plane. In the first lesson, "Descartes Was Really Smart," you will get to know the Cartesian Plane, measure distance in it, and find the equations of lines. The second lesson introduces the idea of a function as an input-output machine, shows you how to graph functions in the Cartesian Plane, and goes over important vocabulary.

Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists must know before moving on to more advanced material. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous rate of change and tangent lines to a curve ~Exponents, logarithms, and the natural log function. ~Probability theory, including Bayes’ theorem. While this course is intended as a general introduction to the math skills needed for data science, it can be considered a prerequisite for learners interested in the course, "Mastering Data Analysis in Excel," which is part of the Excel to MySQL Data Science Specialization. Learners who master Data Science Math Skills will be fully prepared for success with the more advanced math concepts introduced in "Mastering Data Analysis in Excel." Good luck and we hope you enjoy the course!

Skills You'll Learn

Bayes' Theorem, Bayesian Probability, Probability, Probability Theory

Taught By

Daniel Egger
Executive in Residence and Director, Center for Quantitative Modeling
Paul Bendich
Assistant research professor of Mathematics; Associate Director for Curricular Engagement at the Information Initiative at Duke

Transcript

Okay, welcome back, everyone.

We're going to start with the first video in our module on functions.

And this is going to be a little idiosyncratic.

We're not going to talk about functions to start,

the way most of you have probably seen it in early school.

You're probably all used to seeing the idea of a function as a graph.

That is if I draw this in the upper left corner here,

your idea of the function is something like that.

Some squiggly line, here's x, here's y and someone says that y = f(x).

We'll get there, that's actually a way to visually depict a function from the real

line to the real line.

But what we're going to do here, and trust me there's a good pedagogical reason for

it, we're going to start a little bit more abstract.

Talk about the general concept of a function from a set to a set.

Just work with me, and then we'll get eventually to the idea of where these

graphs come from and what they have to do with what I'm talking about here.

So here's a picture I want you to have in mind.

Supposed you have a set A, it's the little bubble on the left, and a set B,

the little bubble on the left.

The general definition is that a function f from A to B

is a rule or a formula or a machine,

which simply transforms every element a and A into some element f(a) and B.

So, really, by machine I mean it.

So here's A, imagine this little cartoon of a journey.

Here's A, little a walking along, gets fed into the machine.

Lord knows what happens in here, we're sort of covering it up,

censoring it with these blue dots, but some crazy stuff happens.

And at the end of the day, out comes an f(a), that's an output over here.

So it's very mechanistic, a is sort of an input, f(a) is an output,

I think you can write that down.

There's the input to the function, there's the output.

And the idea is the function is the machine, it's not a graph.

It's not any of the individual input output.

It is the rule which transforms one to the other.

Okay, great, that's enough talking.

Let's get some examples.

Let's take a really, really simple example of a set.

Suppose our set A consists of 1, and 2, and 10.

And suppose our set B consists of Apple,

Daniel Egger, which we'll call DE, and Monkey.

And we might define a function f from A to B.

And really here you should have the absolute freedom to do whatever

you want with the function.

You might say that f(1) = Apple.

You might say that f(2) = Apple.

And you might say that f(10) = Monkey.

So to give you that in pictures, here's 1, here's 2, here's 10.

Here's Apple, here's Daniel Egger, here's Monkey.

And f sends 1 to Apple, 2 to Apple, and 10 to Monkey.

That's it, that's a function, it's a machine that does this.

Why it does that we don't really know.

Okay, the short of this isn't just a little gross abstract concept.

Let me recast one of the things we did before in functional language.

Suppose for example, that x is equal to all the people in the VBS study.

That's the third time we've mentioned it.

It may actually be a real disease, see if we can get a patent on the treatment.

All people in the VBS study.

And suppose y consists of two symbols, plus for positive, minus for negative.

And let's define a function called test from x to y.

This function is the medical test that we take when we want to tell whether or

not you have VBS.

All we're going to do here is say that test for the particular person.

If we write the test of a person equals plus,

that means that person tested positive.

If we write that test of a person equals minus,

that means that person tested negative.

And so it's just a way of operationalizing the idea, it's that function.

Let's give another example.

Suppose you're running a business.

And let's say that capital Y stands for all the years.

So this might be going from 2010, 2011, 2012, on forever, and over here

on the other side we have the real number line.

Define a function called Profit.

From Y to R.

Where profit of a particular year

is equal to the profit in that year, profit / loss in that year.

You can see here why the target of the function,

the real line, is the real line, because you might want positive or negative.

It might be that profit

In 2011 was $1,007 that might be the profit in 2012 was -10,000.

This might be a typical thesis on running.

We're not doing that well for example.

So that's the basic concept.

There's really not much more to it that's a function from a set to a set.

Okay, so let's get a little bit more advanced here and

tie into something that you see a lot in machine learning.

The dirty truth is that you don't actually have functions in life, usually.

You don't know what every input to output is.

Very often what you do in what's called supervised learning

is figure out functions from a little bit of examples of input and output.

So, for example, the profit function.

From years to the real line, you actually knew what that was.

You knew what every output would be for every input.

If you know what profit of the year was for every year, you'd be in business,

as they say.

If you knew what the result of test was on every person,

you wouldn't have to give them the test.

Very often what you do in Supervised Learning,

is you're often given some examples.

So for example if you're trying to figure out you have a set A and a set B and

you have a mystery function f and you're trying to figure out.

You're often given some examples of a and

A and outputs f(a) and B.

So you might be given some people in their test results,

you might be given some years of the profit in your mission.

Figure out the function, to figure out a trend, a pattern,

some call trend analysis, pattern analysis.

Not getting into the details, it's a really really big rabbit hole,

but the whole field of supervised learning is literally about that.

You're given examples of input output pairs, try to figure out the function.

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