3 Congruence

... During the course of the proof of theorem 3.10 , we proved the following useful result. Theorem 3.12 If ab ≡ ac mod n and if gcd(a, n) = 1, then we have b ≡ c mod n. In short, we can cancel the factor a from both sides of the congruence so long as gcd(a, n) = 1. In algebra, we learn that we “can div ...

... During the course of the proof of theorem 3.10 , we proved the following useful result. Theorem 3.12 If ab ≡ ac mod n and if gcd(a, n) = 1, then we have b ≡ c mod n. In short, we can cancel the factor a from both sides of the congruence so long as gcd(a, n) = 1. In algebra, we learn that we “can div ...

Diskrete Mathematik für Informatik (SS 2017)

... I hope that these slides will serve as a practice-minded introduction to various aspects of discrete mathematics which are of importance for computer science. I would like to warn you explicitly not to regard these slides as the sole source of information on the topics of my course. It may and will ...

... I hope that these slides will serve as a practice-minded introduction to various aspects of discrete mathematics which are of importance for computer science. I would like to warn you explicitly not to regard these slides as the sole source of information on the topics of my course. It may and will ...

Analysis Notes (only a draft, and the first one!)

... We will be interested just in two (binary) operations on R, called addition and multiplication. Apart from these two operations, we will also be interested in a (binary) relation <. Our definition will take some time, till page 23. Definition 2.0.1 A set R together with two binary operations + and × ...

... We will be interested just in two (binary) operations on R, called addition and multiplication. Apart from these two operations, we will also be interested in a (binary) relation <. Our definition will take some time, till page 23. Definition 2.0.1 A set R together with two binary operations + and × ...

# 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.