1. 1 Number Theory and Methods of Proof Content: Properties of integer, rational and real numbers. Underlying theme: Methods of mathematical proofs.

2. 2 This Lecture Even and odd integers Prime and composite integers Constructive and nonconstructive proofs Method of direct proof

3. 3 Even and Odd Integers  Definition: An integer n ● is even iff  an integer k such that n=2k; ● is odd iff  an integer k such that n=2k+1. Ex: If x and y are integers, is even or odd?

4. 4 Prime and Composite Integers  Definition: An integer n (which is >1) ● is prime iff  positive integers r and s, if n=r·s then r=1 or s=1; ● is composite iff  positive integers r and s such that n=r·s and r≠1 and s≠1.  Examples: 5, 7, 23 are prime; 4, 22, 16042 are composite.

5. 5 Proving Existential Statements  Existential statement:  x  D such that P(x)  2 proof methods for existential statements: ● Constructive proofs; ● Nonconstructive proofs.

6. 6 Constructive Proofs of Existence  2 ways: 1) Find x that makes P(x) true. 2) Give a set of directions for finding such x.  Example: 1) There are integer numbers a,b and c such that Proof: For a=3, b=4 and c=5,

7. 7 Constructive Proofs of Existence 2) Suppose a,b  Z such that 1<a<b. Prove that there is a composite even integer c such that a 2 <c<b 2. Proof: By division into cases: (a) Suppose a is even. Then a=2k for some integer k. (by definition) Hence c=a·b = (2k)·b = 2·(k·b) is even integer; (because k·b is an integer) c=a·b is composite (because a≠1 and b≠1); c=a·b>a 2 (because a<b). c=a·b<b 2 (because a<b). (b) Suppose b is even. (c) Suppose both a and b are odd.

8. 8 Nonconstructive Proofs of Existence  2 ways: (1) Show that the existence of x is guaranteed by an axiom or a previously proved theorem. (2) Show that the assumption that there is no such x leads to a contradiction.  Disadvantage: Often these methods give no clue how to find x.

9. 9 Proving Universal Statements  Universal statement:  x  D if P(x) then Q(x)  Proof methods for universal statements: ● Method of exhaustion; ● Method of generalizing from the generic particular. (show the property for a particular but arbitrarily chosen x)

10. 10 Method of Direct Proof  The statement:  x  D if P(x) then Q(x). Suppose x is a particular but arbitrarily chosen element of D for which P(x) is true; Show the conclusion Q(x) is true by using ♦ definitions; ♦ previously established results; ♦ rules of logical inference.

11. Method of Direct Proof (Ex.)  Show  x  Z if x is odd then 3x+9 is even. Proof: Suppose x is an arbitrarily chosen odd integer. Then x=2k+1 for some integer k. (by definition) So 3x+9 = 3(2k+1)+9 (by substitution) = 6k+3+9 (by distributive law) = 2(3k+6) (by factoring out a 2) (*) 3k+6 is an integer. (**) Hence 3x+9 is even based on (*), (**) and definition of even integers. ▀ (this is what we needed to show)

12. 12 Directions for writing proofs 1)Write the theorem to be proved. 2)Clearly mark the beginning of your proof with the word Proof. 3) Make your proof self-contained. (Identify all variables used in the proof; state the sources of outside facts). 4) Write proofs in complete English sentences.