{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ln1-MIT

ln1-MIT - Massachusetts Institute of Technology...

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

Massachusetts Institute of Technology Course Notes 1± 6.042J/18.062J, Fall ’02 : Mathematics for Computer Science± Professor Albert Meyer and Dr. Radhika Nagpal± Proofs± 1 What is a Proof? A proof is a method of ascertaining truth. There are many ways to do this: Jury Trial Truth is ascertained by twelve people selected at random. Word of God Truth is ascertained by communication with God, perhaps via a third party. Word of Boss Truth is ascertained from someone with whom it is unwise to disagree. Experimental Science The truth is guessed and the hypothesis is confirmed or refuted by experi- ments. Sampling The truth is obtained by statistical analysis of many bits of evidence. For example, public opinion is obtained by polling only a representative sample. Inner Conviction/Mysticism My program is perfect. I know this to be true.” “I don’t see why not...” Claim something is true and then shift the burden of proof to anyone who disagrees with you. “Cogito ergo sum” Proof by reasoning about undefined terms. This Latin quote translates as “I think, therefore I am.” It comes from the beginning of a famous essay by the 17th century Mathematician/Philospher, Rene ´ Descartes. It may be one of the most famous quotes in the world: do a web search on the phrase and you will be ﬂooded with hits. Deducing your existence from the fact that you’re thinking about your existence sounds like a pretty cool starting axiom. But it ain’t Math. In fact, Descartes goes on shortly to conclude that there is an infinitely beneficent God. Mathematics also has a specific notion of “proof” or way of ascertaining truth. Definition. A formal proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms . The three key ideas in this definition are highlighted: proposition, logical deduction, and axiom. Each of these terms is discussed in a section below. Copyright © 2002, Prof. Albert R. Meyer.

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

View Full Document
2 Course Notes 1: Proofs 2 Propositions Definition. A proposition is a statement that is either true or false. This definition sounds very general, but it does exclude sentences such as, “Wherefore art thou Romeo?” and “Give me an A!”. Proposition 2.1. 2 + 3 = 5. This proposition is true. Proposition 2.2. Let p ( n ) ::= n 2 + n + 41 . n N p ( n ) is a prime number . The symbol is read “for all”. The symbol N stands for the set of natural numbers , which are 0, 1, 2, 3, . . . ; (ask your TA for the complete list). A prime is a natural number greater than one that is not divisible by any other natural number other than 1 and itself, for example, 2, 3, 5, 7, 11, . . . . Let’s try some numerical experimentation to check this proposition: p (0) = 41 which is prime. p (1) = 43 which is prime. p (2) = 47 which is prime. p (3) = 53 which is prime. . . . p (20) = 461 which is prime. Hmmm, starts to look like a plausible claim. In fact we can keep checking through n = 39 and confirm that p (39) = 1601 is prime.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

ln1-MIT - Massachusetts Institute of Technology...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online