{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

3.1N_Proving_Universal_Statements

# 3.1N_Proving_Universal_Statements - Math 280 Strategies of...

This preview shows pages 1–3. Sign up to view the full content.

Math 280 Strategies of Proof Fall 2007 Class Notes 3.1 Proving Universal Statements The method of proof used to prove a mathematical statement is determined by the statement’s logical structure. We can determine the logical structure by writing the statement in symbolic form, so this is the first step in the proof of any mathematical statement. Almost all of the statements which you’ll be concerned with proving in your mathematics courses will involve one or more quantifiers. So we want to begin our examination of proof techniques by discussing how to prove the general universal statement and how to prove the general existential statement. The overwhelming majority of mathematical theorem statements are universally quantified statements of the general form x p ( x ), where the propositional function p ( x ) may be either simple or compound. We may also have more than one variable involved. The universal statement x p ( x ) is true provided that the propositional function p ( x ) is true for all elements x in the domain D . So to prove the statement x p ( x ) is true, we need to show that, for each element x in the domain D , the propositional function p ( x ) is true. There are two basic approaches to proving a universal statement: proof by exhaustion and proof by arbitrary element. Proof by Exhaustion When the domain D is a finite set, we can prove that x p ( x ) is true by systematically checking that p ( x ) is true for each x D . This method of proof is called proof by exhaustion . Here are two examples of proofs by exhaustion. Example. Prove: For all positive integers n with 1 n 10, n 2 n + 11 is a prime number. Solution. Define the propositional function p ( n ) : n 2 n + 11 is a prime number. The statement has symbolic form n p ( n ) , where the domain for n is the set D = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . Proof. For a proof by exhaustion of n p ( n ), we need to show that p ( n ) is true for each value of n in the domain D . Substituting each value of n into the propositional function, we have p (1) : 1 2 1 + 11 = 11 is a prime number. p (2) : 2 2 2 + 11 = 13 is a prime number. p (3) : 3 2 3 + 11 = 17 is a prime number. p (4) : 4 2 4 + 11 = 23 is a prime number. p (5) : 5 2 5 + 11 = 31 is a prime number. p (6) : 6 2 6 + 11 = 41 is a prime number. p (7) : 7 2 7 + 11 = 53 is a prime number. p (8) : 8 2 8 + 11 = 67 is a prime number. p (9) : 9 2 9 + 11 = 83 is a prime number. p (10) : 10 2 10 + 11 = 101 is a prime number. Since each of these statements is true. then n p ( n ) is true, which proves this statement. 55

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
56 Chapter 3 Proof Techniques Example. Let α be an irrational number. A rational number a 0 /b 0 is called a good approximation to α provided that for every rational number a/b with 1 b < b 0 , Ø Ø Ø Ø α a 0 b 0 Ø Ø Ø Ø < Ø Ø Ø Ø α a b Ø Ø Ø Ø . This condition means a 0 /b 0 is closer to α than every other rational number with a smaller denom- inator. We assume all rational numbers are in lowest terms.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern