Chapter 3. Mathematical Reasoning 3.1 Methods of proof A theorem is a statement that can be shown to be true. A proof is to demonstrate that a theorem is true with a sequence of statements that form an argument. An axiom or postulate is the underlying assumption about mathematical structures, the hypothesis of the theorem to be proved, and previously proved theorems. The rules of inference are the means used to draw conclusion from other assertions which tie together the steps of a proof. A lemma is a simple theorem used in the proof of other theorems. A corollary is a proposition that can be established directly from a theorem that has been proved. A conjecture is a statement whose truth is unknown. When its proof is found, it becomes a theorem.

Rules of Inference The rules of inference provide the justification of the steps used to show that a conclusion follows logically from a set of hypotheses. Basis of the rule of inference: the law of detachment conclusion hypotheses Example 1 It is snowing today. If it snows today, then we will go skiing. We will go skiing.

Rules of InferenceTautologyName Addition Simplification Conjunction Modus ponens (detachment law) Modus tollens Hypothetical syllogism Disjunctive syllogism Table 1 Rules of Inference

Example 2 Example 3 Solution Example 4 Solution

Example 5 Solution Example 6 Show the following hypotheses : (1) It is not sunny this afternoon and it is colder than yesterday. (2)We will go swimming only if it is sunny. (3) If we do not go swimming, then we will take a canoe trip. (4) If we take a canoe trip, then we will be home by sunset. lead to the conclusion “We will be home by sunset”. p: It is sunny this afternoon. q r s t

Example 7 Show that the hypotheses (1)If you send me an message, then I will finish writing the program, (2) If you do not send me an massage, then I will go to sleep early, (3) If I go to sleep early, then I will wake up feeling refreshed lead to the conclusion “If I do not finish writing program, then I will wake up feeling refreshed”. p q r s

3.2 Mathematical Induction How to prove 1+2+…+n=n(n+1)/2 for n=1,2,…? The Well-Ordering Property Every nonempty set of nonnegative integers has a least element. Use mathematical induction! Mathematical Induction(1) Mathematical induction is used to prove the form Where the universe of discourse is the set of positive integers. 1. Basis step. The proposition P(1) is shown to be true. 2.Inductive step. The implication

P(1) Why does Mathematical induction work? Law of detachment Example 1 P(n)

Example 2 Use Mathematical induction to prove the inequality P(n)

Example 3 P(n) Use Mathematical induction to prove that Mathematical Induction(2) 1. Basis step. The proposition P(m) is shown to be true. 2.Inductive step. The implication Mathematical induction is used to prove the form Where the universe of discourse is the set of contiguous integers: m,m+1,m+2,….

Example 4 Sums of Geometric Progressions P(n) Use Mathematical induction to prove the following formula:

Use Mathematical induction to prove thatP(n) Example 5 Inequality for Harmonic Numbers.

Example 6 The numbers of Subsets of a Finite Set P(n) Use Mathematical induction to prove that X X

3.3 Recursive Definitions Recursive definitions: defining an object by itself. Recursively defined functions To define a function with these of nonnegative integers as its domain, 1. Specify the value of the function at zero. 2. Give a rule for finding its value at an integer from its values at smaller integers. Such a definition is called a recursive or inductive definition. Example 1 Function f is recursively defined by f(0)=3, f(n+1)=2f(n)+3 ． Find f(1),f(2) and f(3). f(1)=2f(0)+3=9, f(2)=2f(1)+3=21, f(3)=2f(3)+3=93.

Example 2 Give an inductive definition of the factorial function F(n)=n!. Solution F(0)=1 F(n+1)=(n+1)F(n) Example 3

Example 4 Example 5