1 Introduction 2 What is Discrete Mathematics?
... logic, basic set theory, proof techniques, number theory, mathematical induction, recursion, recurrence relations, counting, probability and graph theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. Some of these applications are lis ...
... logic, basic set theory, proof techniques, number theory, mathematical induction, recursion, recurrence relations, counting, probability and graph theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. Some of these applications are lis ...
Gödel`s First Incompleteness Theorem
... axiomatisation of number theory. That is, any such axiomatisation will either yield a proof for some false statement or will fail to yield a proof for some true one. ...
... axiomatisation of number theory. That is, any such axiomatisation will either yield a proof for some false statement or will fail to yield a proof for some true one. ...
In Class Slides
... assumed, established, or is still to be deduced. • If it is assumed use words like “Suppose” or “Assume” • If it is still to be shown use “We must show that” ...
... assumed, established, or is still to be deduced. • If it is assumed use words like “Suppose” or “Assume” • If it is still to be shown use “We must show that” ...
Chapter 1: The Foundations: Logic and Proofs
... Assume x is not even and show x2 is not even. If x is not even then it must be odd. So, x = 2k + 1 for some k. Then x2 = (2k+1) 2 = 2(2k 2+2k)+1 which is odd and hence not even. This completes the proof of the second case. Therefore we have shown x is even iff x2 is even. Since x was arbitrary, the ...
... Assume x is not even and show x2 is not even. If x is not even then it must be odd. So, x = 2k + 1 for some k. Then x2 = (2k+1) 2 = 2(2k 2+2k)+1 which is odd and hence not even. This completes the proof of the second case. Therefore we have shown x is even iff x2 is even. Since x was arbitrary, the ...
If…then statements If A then B The if…then statements is a
... If…then statements If A then B The if…then statements is a conditional statement. B “requires” A to happen. This establishes a conditional relationship. A proof is a formal justification of a hypothesis. A hypothesis is a possible explanation for an observation that has yet to be established as fact ...
... If…then statements If A then B The if…then statements is a conditional statement. B “requires” A to happen. This establishes a conditional relationship. A proof is a formal justification of a hypothesis. A hypothesis is a possible explanation for an observation that has yet to be established as fact ...
R : M T
... Assume, for contradiction, the opposite of the statement you’re trying to prove. Then do stuff to reach a contradiction. Conclude that your assumption must be false after all. • Proof by Induction Base case: Prove the statement is true for n=1 Inductive hypothesis: Assume that the statement is true ...
... Assume, for contradiction, the opposite of the statement you’re trying to prove. Then do stuff to reach a contradiction. Conclude that your assumption must be false after all. • Proof by Induction Base case: Prove the statement is true for n=1 Inductive hypothesis: Assume that the statement is true ...
ANALYSIS I A Number Called e
... These supplementary notes by H A Priestley provide a (non-examinable) proof of the useful fact that ...
... These supplementary notes by H A Priestley provide a (non-examinable) proof of the useful fact that ...
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.