MATH 224 – Discrete Mathematics

1. MATH 224 – Discrete Mathematics Why Study Discrete Math • Determination of the efficiency of algorithms, e.g., insertion sort versus selection sort. Can you provide an example? • Boolean expressions for controlling loops and conditional statements are based on the propositional calculus and Boolean algebra. What is an example in C++? • The building blocks of computers – logic gates implement Boolean expressions. • Design of programs and algorithms is similar to developing mathematical proofs. • Conversion of high level languages to machine code makes use of formal language theory. Name some high level languages. What is machine code? • Induction and recursion are the basis for algorithms and programs that use repeated instructions, e.g., for statements, while statements and recursive functions.

2. MATH 224 – Discrete Mathematics Why Study Discrete Math Continued • Security including encryption and authentication make use of math concepts. Can you give an example of where encryption is used? What is authentication? • Many data structures make use of trees, e.g., heaps, binary search trees, databases. • The theory behind caching and paging uses mathematics. Do you know what these are? • Graph theory is used in communication networks, artificial intelligence, computer games, computer animation among others. • And much much more. • Perquisite for CS 340 − Required for ABET accreditation

3. MATH 224 – Discrete Mathematics Propositions Propositions correspond to Boolean expressions in C++. They are statements, sometimes represented by variables, that may be either true or false. Examples of proposition like expressions (actually conditional statements) in C++ include: • X <= Y • Z > 10 && Z <= 20 • !(Z <= 10 || Z > 20) • Flag -- where Flag is a boolean variable • !Flag || X % 2 == 0 -- When is X % 2 == 0 ?

4. MATH 224 – Discrete Mathematics Common Logical (connectives) Operators Note that the conditional is not quite the same as the ifstatement in C++. In programming languages the condition is not a proposition but a statement about what code should be executed based on whether or not a Boolean expression is either true or false. Can you give an example of a conditional statement in C++? How can the other operators (→, ↔ and + ) be expressed in C++?

5. MATH 224 – Discrete Mathematics The Meaning of Logical Operators Using Truth Tables These are called compound propositions or propositional clauses. Can you create a similar table with one or both of P replaced by ¬P and Q replaced by ¬Q? Bitwise operators in C++ among others use & (and) | (or) and ^ (xor). These are not the same as the logical operators. Read the text on this topic (pages 15 and 16) and be prepared to answer questions.

6. MATH 224 – Discrete Mathematics Using Truth Tables More complex clauses may be created using multiple truth connectives. In the truth table below the third and fourth columns show the truth values for two clauses with a single operator and then the result when these two clauses are combined with a third operator. What would the truth values be if the left and right side of the conditional were to be reversed? .

7. MATH 224 – Discrete Mathematics Contrapositive On page 8 of our textbook there is a discussion of the terms converse, contrapositive and inverse. The contrapositive is the one most commonly used in proofs since it has the same truth value as the original proposition. The contrapositive of p → q is ¬q → ¬p. For example consider : If x = 2 then the √x is irrational (equivalent to p → q). The contrapositive would be : if the √x is rational then x ≠ 2 (¬q → ¬p) Assume that √x = a/b a rational number in reduced form. Then since a2/b2= x, x ≠ 2. Note that a and b have no common factors unless b= 1. Thus the original proposition is true. Why is b≠ 1? Can you prove for all integers x that the √x is either an integer or irrational? .

8. MATH 224 – Discrete Mathematics Logic Puzzles Example 18 (pages 13-14): Knights are always truthful and knaves always lie. A says B is a Knight and B says we are different. If A is a knight then B must also be a knight. But that leads to a contraction since they must be different. If A is a knave then B must be a knave since knaves lie. B says they are different which would be a lie if B is a knave. Thus the two statements are consistent. Both A and B are knaves. Problem 43 (page 19): Given statements 1 through 100, “Exactly n statements are false.” Which statement must be true? If instead we have, “at least n statements are false.” Which statements are true? What would be the case if there are 99 statements instead of 100? Note that it is possible to have a statement that either contradicts itself or a group of statements that are contradictory. This is called a paradox.

9. MATH 224 – Discrete Mathematics More Logic Puzzles Problem 46 (pages 20): Each member of a group of cannibals either lies or does not lie. An explorer must determine if a given cannibal tells the truth or not. Why does asking him if he is a liar not help? What yes or no question should the explorer ask? How would you go about analyzing this type of problem? One of the tricks that often works is the use of a double negative, i.e., the negation of a negation.

10. MATH 224 – Discrete Mathematics More Logic Puzzles Dr. Who was a British television program from the 1960s and 70s. He was an explorer who used an old telephone booth as his spaceship. In one episode he had to choose between two doors. One door led to freedom and the other to his certain death. There were two robots in the room. One always lied and one always told the truth. What single yes or noquestion should Dr. Who ask of one of the robots in order to determine which door to take?

11. MATH 224 – Discrete Mathematics More Logic Puzzles Evaluating a While Statement A jar contains both white and black beans. You take two beans from the jar. If both of them are black you return one black bean to the jar. If one is black and one is white you return the white bean. If both are white you take both of them and add a black bean to the jar. You repeat this step until you are tired or until the jar has only one bean in it. Will this process ever stop? How many beans will be left at the end? Can you say anything about the original number of black and white beans based on the beans that are left? These properties are called preconditions. Are there any invariant properties?

12. MATH 224 – Discrete Mathematics Semi-formal Proofs By using propositional clauses that have already been shown to be true it is possible to derive equivalent propositions or to prove that a proposition is a tautology (always true). Below is an example of a rather trivial derivation.

13. MATH 224 – Discrete Mathematics Semi-formal Proofs Here is a more complex derivation. Using p ↔ q ≡ (p → q) ^ (q → p) show that p ↔ q ≡ (p ^ q) v (¬p ^ ¬q) p ↔ q ≡ (p → q) ^ (q → p) Given ≡ (¬p V q) ^ (¬q V p) Table 7, Page 25 ≡ (¬p ^ ¬q)v(¬p ^ p)v(¬q ^ q)v(q ^ p) distributive laws Table 6, Page 24 ≡ (¬p ^ ¬q) v F v F v (q ^ p) negation law, Table 6 ≡ (p ^ q) v (¬q ^ ¬p) identity law, Table 6 ≡ (p ^ q) v (¬p ^ ¬q) commutative law, Table 6 Is (p ↔ q) ↔ [(p ^ q) v (¬p ^ ¬q)] a tautology? Explain your answer.