Introduction

Mathematics is often split into two broad styles. Continuous mathematics works with the set of real numbers, which is uncountably infinite -- between any two real numbers there is always another whole uncountable set of numbers, so a continuous function can be sketched as a smooth, unbroken curve. Discrete mathematics, by contrast, works with distinct values that are either finite or countably infinite -- between any two points there are only finitely many (or countably many) points, and a function on a finite set can be listed completely as a set of ordered pairs.

The mathematicians of the late 19th and early 20th centuries built discrete mathematics around sets like $\mathbb N$ that are either finite or countably infinite -- called discrete sets. The defining feature of a discrete set is that a one-to-one correspondence can always be set up between it and (a subset of) the natural numbers, so its elements can be listed as a sequence -- something impossible for an uncountable set like $\mathbb R$, whose points are packed together with no gaps at all.

Why discrete mathematics matters today. Computers work natively with discrete, countable data, so computer science is, in large part, the science of describing and reasoning about discrete sets precisely and concisely -- and modern programming languages are built to express exactly these discrete descriptions. Studying discrete mathematics sharpens general reasoning and problem-solving, independent of any one application.

Branches of discrete mathematics include combinatorics, mathematical logic, Boolean algebra, graph theory and coding theory. Three of these -- permutations, combinations, and mathematical induction -- were already studied in Class 11. This chapter takes up two more: binary operations (Section 12.2) and mathematical logic (Section 12.3).