Discrete Mathematics

1. Discrete Mathematics • Discrete means • apart • distinct • away from each other • not continuous Examples: The (sound) pitch of a violin is continuous. The (sound) pitch of a piano is discrete.

2. An electric analogue clock shows continuous time. A digital clock shows discrete time. A conventional photograph shows almost continuous shades of color. A digital picture shows discrete shades of color

3. -10 -5 0 5 10 Subsets of the Real numbers A subset of real numbers is said to be discrete if every member in the subset has a neighborhood that separates it from all other members in the subset. For example, the above set (indicated by red dots) is discrete (click to see the “private” neighborhoods. On the other hand, the set of fractions between 0 and 1 is not discrete.

4. Subsets of the Real numbers It is clear that every finite subset of the real number is discrete. But there are infinite discrete subsets as well, such as the set of integers. Therefore, finite mathematics is a part of discrete mathematics. But discrete mathematics contains more than just finite mathematics.

5. Some characteristics of Discrete mathematics • In discrete mathematics, we do not take limit as x approaches a certain number. • We do not use approximation techniques to solve equations.

6. Whydo we study discrete mathematics? • there are lots of intrinsically discrete problems that cannot be solved by “calculus” type techniques. • computers (in the present) can only handle discrete structures. • a discrete model can be used to approximate a continuous model to a very high degree of accuracy. (such as digital photos and digital music)

7. Chapter 1 The Logic of Compound Statements

8. Mathematical Logic In order to study discrete mathematics and understand computer programming, one must have some basic knowledge of mathematical logic. Hence in any beginning course of discrete mathematics, about 50% of the time is spent on mathematical logic.

9. Mathematical Logic • A set of precise rules that governs the operations of computers (and our mind). Propositional Calculus Predicate Calculus • A proposition is a sentence that is either true or false but not both.(In particular, it cannot be a question.) • Examples: • 2 + 2 = 5 • sin(π/6) = 0.5 • A predicate is a sentence that contains variables, and when the variables are substituted by numbers or actual objects, it becomes a proposition. • Examples: • x > 4 • a2 + b2 = c2

10. Predicate Calculus Propositional Calculus A proposition cannot have free variables. Propositional calculus is analogous to Arithmetic where we do not deal with variables Predicate calculus on the other hand is analogous to Algebra, which is more complex than arithmetic but it requires the knowledge of arithmetic. Note: a proposition is also called a statement.

11. More examples of Propositions Determine whether each of the following is a proposition: Yes. 1. Washington, D.C., is the capital of USA. No. 2. Please read this carefully. Yes. 3. All Martians like pepperoni on their pizza. Yes. 4. Jane forgot to bring her umbrella. 5. Would you please pass me the salt? No.