(A B) (A B) (A B) (A B)
... O(g(x)) if there are constants CN and kR such that |f(x)| C|g(x)| whenever x > k. • We say “f(x) is big-oh of g(x)”. • The intuitive meaning is that as x gets large, the values of f(x) are no larger than a constant time the values of g(x), or f(x) is growing no faster than g(x). • The suppositio ...
... O(g(x)) if there are constants CN and kR such that |f(x)| C|g(x)| whenever x > k. • We say “f(x) is big-oh of g(x)”. • The intuitive meaning is that as x gets large, the values of f(x) are no larger than a constant time the values of g(x), or f(x) is growing no faster than g(x). • The suppositio ...
Basic Logic and Fregean Set Theory - MSCS
... of constructive logic, only failing in cases of the kind mentioned above. In areas like computer algebra constructive logic may perform relatively more prominent functions. The idea of using models of nature with a logic different from the classical one is not new. Quantum logic has been used to mo ...
... of constructive logic, only failing in cases of the kind mentioned above. In areas like computer algebra constructive logic may perform relatively more prominent functions. The idea of using models of nature with a logic different from the classical one is not new. Quantum logic has been used to mo ...
Lecture 3
... To work with objects such as 2 in a systematic way we need to consider a much larger system of numbers - the real numbers R. To begin with we’ll work with a heuristic idea of R as the set of all numbers which can be represented by an infinite decimal expansion. It therefore corresponds to our intuit ...
... To work with objects such as 2 in a systematic way we need to consider a much larger system of numbers - the real numbers R. To begin with we’ll work with a heuristic idea of R as the set of all numbers which can be represented by an infinite decimal expansion. It therefore corresponds to our intuit ...
Pythagorean Triples and Fermat`s Last Theorem
... We conjecture that c − b and c + b are always squares of odd numbers. How can we prove this? First, we observe that c − b and c + b seem to have no common factors. Suppose that d is a common factor of c − b and c + b; that is, d divides both c − b and c + b. It is an exercise to show that d also div ...
... We conjecture that c − b and c + b are always squares of odd numbers. How can we prove this? First, we observe that c − b and c + b seem to have no common factors. Suppose that d is a common factor of c − b and c + b; that is, d divides both c − b and c + b. It is an exercise to show that d also div ...
Natural numbers Math 122 Calculus III
... The natural numbers, N, and mathematical induction. These are also called positive integers, or whole positive numbers. They include 1, 2, 3, and so forth. We’ll use the notation N for the set of all natural numbers. A serious study of the foundations of mathematics would study N in detail. We won’t ...
... The natural numbers, N, and mathematical induction. These are also called positive integers, or whole positive numbers. They include 1, 2, 3, and so forth. We’ll use the notation N for the set of all natural numbers. A serious study of the foundations of mathematics would study N in detail. We won’t ...
Math Review
... N=1+(P1P2…Pk). Clearly N > Pk so it is not prime. But N is not divisible by P1, P2, P3, P4, …, or Pk (remainder of 1). Contradiction, because every number is divisible by a prime (we just proved it by induction)! So assumption of false was wrong. In other words, theorem is true. ...
... N=1+(P1P2…Pk). Clearly N > Pk so it is not prime. But N is not divisible by P1, P2, P3, P4, …, or Pk (remainder of 1). Contradiction, because every number is divisible by a prime (we just proved it by induction)! So assumption of false was wrong. In other words, theorem is true. ...
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.