24.1. Functions

So, what is a function? So, imagine you are given two sets. Set A and a set B and when I say I have a function, say, f: A → B. Then it is a special type of relation from the set A to the set B.

A function is a specific relationship between two sets, where each element from the first set (set A) is associated with exactly one element from the second set (set B). This implies that if you pick any item from set A, you can find only one corresponding item in set B. Functions are often written in the form f: A → B, where 'f' represents the function itself, indicating the mapping from A to B.

Think of set A as a group of students and set B as a group of lockers. Each student (element in set A) is assigned to one specific locker (element in set B). Each locker can hold only one student’s belongings, so every student has their unique locker. In this case, the function represents the assignment of lockers to students.

And what is the specialty about this relation? The specialty here is that, each element of the set A is assigned exactly one element of the set B.

This is the defining characteristic of a function: for every input from set A, there is a unique output in set B. No element in A can point to multiple elements in B, ensuring a one-to-one mapping.

Imagine a teacher assigning grades to students. Each student (element of set A) receives one specific grade (element of set B). No student can receive multiple grades for the same assignment, reflecting the one-to-one nature of a function.

We also use the term domain and the co-domain in the context of a function. The set A will be called as the domain of the function while B is called as the co-domain of the function.

Every function has a domain, which is the set of all possible inputs (in this case, set A), and a co-domain, which is the set of potential outputs (set B). Understanding these terms helps clarify what values the function can accept and what values it can produce.

Consider a vending machine (the function). The domain consists of the coins (inputs) you can insert—these are the values the machine accepts. The co-domain consists of the snacks (outputs) that can be dispensed. Some coins may not lead to any snack if not enough coins are inserted; hence, some snacks may never be dispensed.

So, now we will be interested to study some important class of functions. So, the first important class of functions is the one to one or injective functions.

An injective function (or one-to-one function) ensures that distinct elements in set A map to distinct elements in set B. If two elements from A have the same corresponding element in B, it contradicts the definition of an injective function. Therefore, for each unique input, there is a unique output.

Imagine a classroom scenario where each student is assigned a unique student ID. If two students share the same ID, it would create confusion. Thus, like an injective function, each student must have their unique ID, representing their distinct identification.

The next important category of function is the onto or surjective functions. If we call it will be called as a surjective function provided the following universal quantification hold.

A surjective function ensures that every element in the co-domain (set B) is covered by at least one element from the domain (set A). So, there can't be any outputs in B that are left without a corresponding input from A.

Think of a delivery service—every address (element in set B) must be served by at least one delivery (element in set A). If some addresses are never reached, the service is not surjective. This guarantees complete coverage of all potential destinations.

The third category of important functions is the one-to-one, onto function. They are also called as bijective functions.

A bijective function possesses both properties: it is injective (one-to-one) and surjective (onto). This means that each element in set A pairs uniquely with an element in set B, and every element in B is paired with an element in A.

Consider a pairing system where each student is assigned to a unique desk, and every desk is occupied by one student. This perfect one-to-one and onto mapping reflects a bijective function—no student shares a desk, and every desk is taken.

Now we will define what we call as the inverse of a function. So, imagine a function f: A → B.

The inverse operation seeks to reverse the function, mapping outputs back to their respective inputs. For a function to have an inverse, it must be bijective, so each element can reliably trace back to its original source without ambiguity.

It’s like deciphering a code. If each code leads back to a unique word (bijective), you can invert the process and retrieve the original word from the code. If some codes were shared (not injective) or some words had no codes (not surjective), deciphering wouldn't be straightforward.

So, now to define the composition of functions. So, imagine you are given two functions, a function g and the function f with appropriate domain and co-domain.

The composition of functions (denoted as f(g(x))) is where the output of one function becomes the input for another. This means that to evaluate the combined function, you first calculate g(x) and then input that result into f.

Consider a scenario where you first buy ingredients (function g) and then make a dish (function f). You cannot prepare the dish without first acquiring the right ingredients. This sequential relationship reflects the composition of functions.

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