Summary
In this chapter, we have discussed the meaning of discrete mathematics and discrete objects. Furthermore, we provided an overview of some of the many applications of discrete mathematics in the real world, especially in the computer and data sciences, which we will discuss in depth in later chapters.
In addition, we have established some common language and notation of importance for discrete mathematics in the form of set notation, which will allow us to refer to mathematical objects with ease, count the size of sets, represent them as Venn diagrams, and much more. Beyond this, we learned about a number of operations that allow us to manipulate sets by combining them, intersecting them, and finding complements. These give rise to some of the foundational results in set theory in De Morgan's laws, which we will make use of in later chapters.
Lastly, we took a look at the ideas of functions and relations, which map mathematical objects such as numbers to one another. While certain types of functions may be familiar to the reader from high school or secondary school, these familiar functions are typically defined on continuous domains. Since we focus on discrete, rather than continuous, sets in discrete mathematics, we drew the distinction between the familiar idea and a new one we need in this field. Similarly, we showed the difference between functions in mathematics and functions in Python and saw that some Python "functions" are mathematical functions, but others are not.
In the remaining four chapters of Part I: Core Concepts of Discrete Mathematics, we will fill our discrete mathematics toolbox with more tools, including logic in Chapter 2, Formal Logic and Constructing Mathematical Proofs, numerical systems, such as binary and decimal, in Chapter 3, Computing with Base n Numbers, counting complex sorts of objects, including permutations and combinations, in Chapter 4, Combinatorics Using SciPy, and dealing with uncertainty and randomness in Chapter 5, Elements of Discrete Probability. With this array of tools, we will be able to consider more and more real-world applications of discrete mathematics.
