Lec11Proofs05
... It is easy to make mistakes, make sure that: 1) All premises pi are true when you prove (p1 AND p2 AND...pn) q 2) Every rule of inference you use is correct. Some proof strategies: To proof pq 1) direct proof: assume p is true, use rules to prove that q is true. 2) indirect proof, assume q is NOT ...
... It is easy to make mistakes, make sure that: 1) All premises pi are true when you prove (p1 AND p2 AND...pn) q 2) Every rule of inference you use is correct. Some proof strategies: To proof pq 1) direct proof: assume p is true, use rules to prove that q is true. 2) indirect proof, assume q is NOT ...
Lec11Proofs
... It is easy to make mistakes, make sure that: 1) All premises pi are true when you prove (p1 AND p2 AND...pn) q 2) Every rule of inference you use is correct. Some proof strategies: To proof pq 1) direct proof: assume p is true, use rules to prove that q is true. 2) indirect proof, assume q is NOT ...
... It is easy to make mistakes, make sure that: 1) All premises pi are true when you prove (p1 AND p2 AND...pn) q 2) Every rule of inference you use is correct. Some proof strategies: To proof pq 1) direct proof: assume p is true, use rules to prove that q is true. 2) indirect proof, assume q is NOT ...
deduction and induction - Singapore Mathematical Society
... This seems to work for any specific value of n that we substitute. Yet how do we prove that it will work for all n ~ 2? Fortunately, there is a method, known as mathematical induction, which establishes a correct inductive guess on a sound basis. In its most elementary application, mathematical indu ...
... This seems to work for any specific value of n that we substitute. Yet how do we prove that it will work for all n ~ 2? Fortunately, there is a method, known as mathematical induction, which establishes a correct inductive guess on a sound basis. In its most elementary application, mathematical indu ...
Math 25 — Solutions to Homework Assignment #4
... 3. A set E of real numbers is called ultra-dense if for every real numbers a < b, the interval (a, b) contains infinitely many elements of E. Prove that E is ultra-dense if and only if it is dense. Proof. Clearly, if E is ultra-dense, then it is also dense. So suppose E is dense in R. Assume there ...
... 3. A set E of real numbers is called ultra-dense if for every real numbers a < b, the interval (a, b) contains infinitely many elements of E. Prove that E is ultra-dense if and only if it is dense. Proof. Clearly, if E is ultra-dense, then it is also dense. So suppose E is dense in R. Assume there ...
Section 2.4 1 Definition of a Limit 2 The Absolute Value Function
... whatever, that is no matter how small we make ε then there is also some small δ, such that when the difference of c and x is within δ then the didderence from f (x) and L is with in ε. This delivers for us the notion of ”near” that I have up to this point not explained. We will be using this definit ...
... whatever, that is no matter how small we make ε then there is also some small δ, such that when the difference of c and x is within δ then the didderence from f (x) and L is with in ε. This delivers for us the notion of ”near” that I have up to this point not explained. We will be using this definit ...
When to Use Indirect Proof
... If 2 were rational then we could write it as 2 = x/y where x and y are integers and y is not 0. By repeatedly cancelling common factors, we can make sure that x and y have no common factors so they are not both even. Then 2 = x2 /y 2 so x2 = 2y 2 so x2 is even. This means x is even, because the squa ...
... If 2 were rational then we could write it as 2 = x/y where x and y are integers and y is not 0. By repeatedly cancelling common factors, we can make sure that x and y have no common factors so they are not both even. Then 2 = x2 /y 2 so x2 = 2y 2 so x2 is even. This means x is even, because the squa ...
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.