We have showed the following sets are countable by constructing a
... ◦ Use the property that if f is O(g) then f+g is O(g) ◦ (i) For nn!+3n+2 +3n100, eliminate 3n+2 and 3n100 since n! grows faster than both of them ◦ (ii) Now simplify nn + n2n, which grows faster? Take the log (base 2) of both (since the log is an increasing function whatever conclusion we draw about ...
... ◦ Use the property that if f is O(g) then f+g is O(g) ◦ (i) For nn!+3n+2 +3n100, eliminate 3n+2 and 3n100 since n! grows faster than both of them ◦ (ii) Now simplify nn + n2n, which grows faster? Take the log (base 2) of both (since the log is an increasing function whatever conclusion we draw about ...
Lecture 6 Induction
... Principle of Mathematical Induction A proposition is defined as a statement that is either true or false. We will at times make a declarative statement as a proposition and then proceed to prove that it is true. Alternately we may provide an example (called a counterexample ) showing that the propos ...
... Principle of Mathematical Induction A proposition is defined as a statement that is either true or false. We will at times make a declarative statement as a proposition and then proceed to prove that it is true. Alternately we may provide an example (called a counterexample ) showing that the propos ...
The Partition Function and Ramanujan`s 5k + 4 Congruence
... Indian mathematician Ramanujan. Ramanujan was inspired by staring at a table of partitions constructed by MacMahon, who found and listed the values for p(n) up to p(200) (see [2]). The congruence is as follows: If a natural number n has the form 5k + 4, where k is a natural number, then p(n) is divi ...
... Indian mathematician Ramanujan. Ramanujan was inspired by staring at a table of partitions constructed by MacMahon, who found and listed the values for p(n) up to p(200) (see [2]). The congruence is as follows: If a natural number n has the form 5k + 4, where k is a natural number, then p(n) is divi ...
Lecture 4: Combinations, Subsets and Multisets
... Change the problem into counting solutions to an equation Let a1 , a2 , . . . , ak be the k different elements of S, each appearing infinitely many times in S. We want to know how many ways there are to choose a1 some number n1 of times, with n1 ≥ 0 a2 some number n2 of times, with n2 ≥ 0 . . . ak s ...
... Change the problem into counting solutions to an equation Let a1 , a2 , . . . , ak be the k different elements of S, each appearing infinitely many times in S. We want to know how many ways there are to choose a1 some number n1 of times, with n1 ≥ 0 a2 some number n2 of times, with n2 ≥ 0 . . . ak s ...
Shimizu.pdf
... A teacher wrote “ 0.999 ” and asked the class, “Besides itself, what does this number equal?” The ensuing discussion revealed four different positions. (1) 0.999 equals 1. It’s just one of those facts I memorized from another class. (2) 0.999 equals 1. But I don’t believe it. (3) I’m convinced 0. ...
... A teacher wrote “ 0.999 ” and asked the class, “Besides itself, what does this number equal?” The ensuing discussion revealed four different positions. (1) 0.999 equals 1. It’s just one of those facts I memorized from another class. (2) 0.999 equals 1. But I don’t believe it. (3) I’m convinced 0. ...
Section 1.3 Predicates and Quantifiers Assume universe of
... Almost nothing in undergraduate and some graduate mathematics courses creates FEAR, as when an instructor announces that the students in his/her class will be required to do proofs. Yet, we all prove statements frequently. We may be required to explain some fact to a child, a high school student or ...
... Almost nothing in undergraduate and some graduate mathematics courses creates FEAR, as when an instructor announces that the students in his/her class will be required to do proofs. Yet, we all prove statements frequently. We may be required to explain some fact to a child, a high school student or ...
Natural deduction
... • After all, we just found this nice method of truth-tables, which can be used to determine the validity or invalidity of any argument in truth-functional logic! • But there are some problems with truth-tables – They can be very inefficient. The reason is that if you have an argument which contains ...
... • After all, we just found this nice method of truth-tables, which can be used to determine the validity or invalidity of any argument in truth-functional logic! • But there are some problems with truth-tables – They can be very inefficient. The reason is that if you have an argument which contains ...
Letter to the Editor
... I'm afraid there was an error in the February issue of The Fibonacci Quarterly. Mr. Shallit's proof that phi is irrational is correct up to the point where he claims that 1/0 can't be an integer. He has no basis for making that claim, as 0 was defined as a rational number, not an integer. The proof ...
... I'm afraid there was an error in the February issue of The Fibonacci Quarterly. Mr. Shallit's proof that phi is irrational is correct up to the point where he claims that 1/0 can't be an integer. He has no basis for making that claim, as 0 was defined as a rational number, not an integer. The proof ...
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.