February 23
... For Tuesdayday: read section 6.1 (and 6.2 if time permits) A shortcut for computing an individual row of Pascal’s triangle: Theorem: C(n,k) = ((n-k+1)/k) C(n,k-1). Application: n=6: 1,6,15,20,15,6,1. Proof: This formula can be rewritten as C(n,k) k = C(n,k-1) (n-k+1). To prove this, note that both s ...
... For Tuesdayday: read section 6.1 (and 6.2 if time permits) A shortcut for computing an individual row of Pascal’s triangle: Theorem: C(n,k) = ((n-k+1)/k) C(n,k-1). Application: n=6: 1,6,15,20,15,6,1. Proof: This formula can be rewritten as C(n,k) k = C(n,k-1) (n-k+1). To prove this, note that both s ...
Lesson 1
... • Inductive Reasoning- the process of arriving at a general conclusion based on observations of specific examples. • We can never be certain that these conclusions are true. • For this reason conclusions are called conjectures, hypotheses, or educated guess. ...
... • Inductive Reasoning- the process of arriving at a general conclusion based on observations of specific examples. • We can never be certain that these conclusions are true. • For this reason conclusions are called conjectures, hypotheses, or educated guess. ...
How to Guess What to Prove Example
... How do you know that 3p1p2p3 = q1q2r1r2. (They do, but it doesn’t follow from the induction hypothesis.) This is a breakdown error. If you’re trying to show something is unique, and you break it down (as we broke down n+1 into r and s) you have to argue that nothing changes if we break it down a dif ...
... How do you know that 3p1p2p3 = q1q2r1r2. (They do, but it doesn’t follow from the induction hypothesis.) This is a breakdown error. If you’re trying to show something is unique, and you break it down (as we broke down n+1 into r and s) you have to argue that nothing changes if we break it down a dif ...
Slides - My E-town
... There are situations where we need more than the fact that the kth element maintains some property to prove the k + 1st element has the same property Strong induction allows us to use the kth element, the k-1st, element, the k-2nd element, and so on This is usually most helpful when the subterms you ...
... There are situations where we need more than the fact that the kth element maintains some property to prove the k + 1st element has the same property Strong induction allows us to use the kth element, the k-1st, element, the k-2nd element, and so on This is usually most helpful when the subterms you ...
Prove
... To prove a statement of the form ∀x, p(x), we prove p(x) in a way that does not depend on x being any particular number, that is, by only using properties known to be true for every number x. Applying a general property/principle to a particular case is called deductive reasoning. logo ...
... To prove a statement of the form ∀x, p(x), we prove p(x) in a way that does not depend on x being any particular number, that is, by only using properties known to be true for every number x. Applying a general property/principle to a particular case is called deductive reasoning. logo ...
Prime Numbers are Infinitely Many: Four Proofs from
... situations and moreover it gives us the opportunity for a deep critical study of considered historical periods. Another important approach is the “voices and echoes” perspective by P. Boero (Boero & Al. 1997 and 1998). 2. Prime numbers are infinitely many In our opinion, the comparison of some diffe ...
... situations and moreover it gives us the opportunity for a deep critical study of considered historical periods. Another important approach is the “voices and echoes” perspective by P. Boero (Boero & Al. 1997 and 1998). 2. Prime numbers are infinitely many In our opinion, the comparison of some diffe ...
Document
... Even in the interval 0,1 there are more real numbers than the number of all natural numbers. However, in this interval there is the same number of reals than the number of all the reals R! Cantor’s diagonal proof: If there were countably many real numbers in the interval 0,1, the numbers could b ...
... Even in the interval 0,1 there are more real numbers than the number of all natural numbers. However, in this interval there is the same number of reals than the number of all the reals R! Cantor’s diagonal proof: If there were countably many real numbers in the interval 0,1, the numbers could b ...
Test 2 Solutions
... This last expression shows that the product is odd, since it is of the form 2n+1, with n = 2ab + a + b. ...
... This last expression shows that the product is odd, since it is of the form 2n+1, with n = 2ab + a + b. ...
Outline for Chapter 10
... This is an outline of the material that I have covered (or will cover) from Chapter 10. It is almost all in the book, but I am skipping around a bit so I thought it would help to collect the relevant material here. I added a few things that I didn’t have time to discuss in the lecture but which are ...
... This is an outline of the material that I have covered (or will cover) from Chapter 10. It is almost all in the book, but I am skipping around a bit so I thought it would help to collect the relevant material here. I added a few things that I didn’t have time to discuss in the lecture but which are ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.