0.2 Discrete structures introduction

Introduction to discrete structures

What is discrete mathematics?

Discrete mathematics is mathematics that deals with discrete objects. Discrete objects are those which are separated from (not connected to/distinct from) each other. Integers (aka whole numbers), rational numbers (ones that can be expressed as the quotient of two integers), automobiles, houses, people etc. are all discrete objects. On the other hand real numbers which include irrational as well as rational numbers are not discrete. As you know between any two different real numbers there is another real number different from either of them. So they are packed without any gaps and can not be separated from their immediate neighbors. In that sense they are not discrete. In this course we will be concerned with objects such as integers, propositions, sets, relations and functions, which are all discrete. We are going to learn concepts associated with them, their properties, and relationships among them among others.

Why discrete mathematics?

Let us first see why we want to be interested in the formal/theoretical approaches in computer science.

Some of the major reasons that we adopt formal approaches are 1) we can handle infinity or large quantity and indefiniteness with them, and 2) results from formal approaches are reusable. As an example, let us consider a simple problem of investment. Suppose that we invest $1,000 every year with expected return of 10% a year. How much are we going to have after 3 years, 5 years, or 10 years? The most naive way to find that out would be the brute force calculation. Let us see what happens to $1,000 invested at the beginning of each year for three years. First let us consider the $1,000 invested at the beginning of the first year. After one year it produces a return of $100. Thus at the beginning of the second year, $1,100, which is equal to $1,000 * (1 + 0.1), is invested. This $1,100 produces $110 at the end of the second year. Thus at the beginning of the third year we have $1,210, which is equal to $1,000 * (1 + 0.1)*(1 + 0.1), or $1,000 * (1 + 0.1)2. After the third year this gives us $1,000 * (1 + 0.1)3. Similarly we can see that the $1,000 invested at the beginning of the second year produces $1,000 * (1 + 0.1)2 at the end of the third year, and the $1,000 invested at the beginning of the third year becomes $1,000 * (1 + 0.1). Thus the total principal and return after three years is $1,000 * (1 + 0.1) + $1,000 * (1 + 0.1)2 + $1,000 * (1 + 0.1)3, which is equal to $3,641.

One can similarly calculate the principal and return for 5 years and for 10 years. It is, however, a long tedious calculation even with calculators. Further, what if you want to know the principal and return for some different returns than 10%, or different periods of time such as 15 years? You would have to do all these calculations all over again. We can avoid these tedious calculations considerably by noting the similarities in these problems and solving them in a more general way. Since all these problems ask for the result of investing a certain amount every year for certain number of years with a certain expected annual return, we use variables, say A, R and n, to represent the principal newly invested every year, the return ratio, and the number of years invested, respectively. With these symbols, the principal and return after n years, denoted by S, can be expressed as S = A(1 + R) + A(1 + R)2 + ... + A(1 + R)n. As well known, this S can be put into a more compact form by first computing S - (1 + R)S as