Functions – Mapping from Sets to Sets
Functions are defined by an input and an output. The input refers to what you are putting into the function, and the output is what comes out of it. Functions can also be described as mapping from one set to another set. The set being mapped from is called the domain, and the set being mapped to is called the range. We will not begin our discussion of functions by examining the most common approach, in which one draws a graph. That method is familiar to you. Instead, we will begin with an approach that’s less well known – one that emphasizes a key property of functions: their ability to transform inputs into outputs. So here is a picture to keep in mind. Suppose we have a set A, represented by the little bubble on the left, and a set B, represented by the little bubble on the left. The general definition of a function f from A to B is a rule or formula that takes every element from A and produces some element from B. For example, we can imagine a machine that takes an apple from A and spits out an orange from B. Here’s A, a little walking along, gets fed into the machine. Lord knows what happens here, we’re sort of covering it up, censoring it with these blue dots. And at the end of the day, out comes an f(a), that’s an output over here. Thus, it is a mechanistic process in which the input a is transformed into an output f(a). The function of a machine is not simply a graph or a list of inputs and outputs. It is the rule that transforms one thing into another. [Graph] Suppose you have a set A that consists of 1, 2 and 10. Suppose your set B has the elements Apple, DE, and Monkey. Given that information, you might define a function f from A to B such that f(1) = Apple, f(2) = Apple and f(10) = Monkey. That is what we mean by a function. To illustrate visually: 1, 2 and 10 are in A; Apple, DE and Monkey are in B; f maps 1 to Apple, 2 to Apple and 10 to Monkey. Such mappings are important in mathematics. They can be used for many purposes including finding patterns and organizing data. [Graph] Let’s recast one of the things. For example, suppose x equals all the people from the VBS study. That’s the third time we’ve mentioned it. It may It actually be a real disease, see if we can get a patent on the treatment. All people from the VBS study. 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. We’re going to say that a test for a person equals plus if that person tested positive and minus if that person tested negative. And so it’s just a way of operationalizing the idea; it’s that function. [Graph] The concept of defining a function is simple. 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. It should be noted that the target of this function- the real line is the real line because it’s possible that profit in 2011 was $1,007 and it might be so that profit in 2012 was -10,000. This might be typical of a thesis on running a business or not doing so well. The main concept we see here is really not much more than defining a function from one set to another set. Y = (. 2010, 2011, 2012, …} TR Profit: Y ->IR Profit(year) – Profit+/loss in that vear Profit(2011) = 1,007 -Profit(2012) = -10.000 We can get a bit more advanced here, and tie into something you see a lot in machine learning. The truth is that you don’t always have functions in life- you often don’t know what every input to output is. So in what’s called supervised learning, you figure out functions from a little bit of examples of input and output. For example: the profit function. From years to the real line, you understood how every output related to 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 the test was on every person, you wouldn’t have to give them the test. What you do in supervised learning is 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. You try to figure out the function that predicts their profit. This is called pattern analysis. Supervised Learning. Profit: Years –> IR Given some examples of a € 4 and outputs f(a) e B. Mission: Figure out f: A–> B. f: A–?>B
