- ScholarWorks@GVSU
... 8. Prove both of the conditional statements: (1) If the area of the right triangle is c 2 =4, then the right triangle is an isosceles triangle. (2) If the right triangle is an isosceles triange, then the area of the right triangle is c 2 =4. ...
... 8. Prove both of the conditional statements: (1) If the area of the right triangle is c 2 =4, then the right triangle is an isosceles triangle. (2) If the right triangle is an isosceles triange, then the area of the right triangle is c 2 =4. ...
Gödel Without (Too Many) Tears
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
Chapter 6
... Calling bad(1) will call bad(1) at line 6 and go into an infinite loop Calling bad(2) will call bad(1) which will once again go into an infinite loop In addition, bad(3), bad(4), and bad(5) all call bad(2) which goes into an infinite loop The only value of n for which this program works is its speci ...
... Calling bad(1) will call bad(1) at line 6 and go into an infinite loop Calling bad(2) will call bad(1) which will once again go into an infinite loop In addition, bad(3), bad(4), and bad(5) all call bad(2) which goes into an infinite loop The only value of n for which this program works is its speci ...
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.