hilbert systems - CSA
... formulas such that each term is either an axiom or follows from earlier terms by one of the rules of inference. A derivation in a Hilbert system from a set S of formulas is a finite sequence Z1, Z2, ..., Zn of formulas such that each term is either an axiom, or is a member of S, or follows from earl ...
... formulas such that each term is either an axiom or follows from earlier terms by one of the rules of inference. A derivation in a Hilbert system from a set S of formulas is a finite sequence Z1, Z2, ..., Zn of formulas such that each term is either an axiom, or is a member of S, or follows from earl ...
Lecture 5
... C) are all obtainable from numbers in Q through a finite sequence of field operations and taking square roots. Theorem. Suppose γ P CzQ. Then there exist α1 , . . . , αn P C with αn “ γ such that rQpα1 , . . . , αi q : Qpα1 , . . . , αi´1 qs “ 2 for i “ 1, . . . , n. In particular, rQpγq : Qs “ 2r f ...
... C) are all obtainable from numbers in Q through a finite sequence of field operations and taking square roots. Theorem. Suppose γ P CzQ. Then there exist α1 , . . . , αn P C with αn “ γ such that rQpα1 , . . . , αi q : Qpα1 , . . . , αi´1 qs “ 2 for i “ 1, . . . , n. In particular, rQpγq : Qs “ 2r f ...
Notes
... there are “holes” in the number line filled√by irrational numbers, such as 2. The Greeks discovered and proved that 2 is not rational. Exercise: find a very simple proof of this. They soon found that the square root of every prime is irrational. By the time of Euclid’s Elements (300 BCE) the irratio ...
... there are “holes” in the number line filled√by irrational numbers, such as 2. The Greeks discovered and proved that 2 is not rational. Exercise: find a very simple proof of this. They soon found that the square root of every prime is irrational. By the time of Euclid’s Elements (300 BCE) the irratio ...
Early_Term_Test Comments
... • Understand set terminology such as set builder notation, and be able to use it. • Know meaning of standard sets like N, Z, Q, R, • Understand standard set theory terminology, e.g., element, subset, cartesian product, universal set, set difference, etc. • Know standard set operations and be able to ...
... • Understand set terminology such as set builder notation, and be able to use it. • Know meaning of standard sets like N, Z, Q, R, • Understand standard set theory terminology, e.g., element, subset, cartesian product, universal set, set difference, etc. • Know standard set operations and be able to ...
Exercise set 2 Number Theory Tuesday SEP 27 2011 at 4 pm. SHARP
... example 10 is irrational. 5. in Euclid’s classical proof, a prime outside {p1 , p2 , . . . , pn } is found by considering prime factors of Nn = p 1 p 2 · · · p n + 1 . Do this beginning with {2}, next being {2, p2 }, where in fact N2 = 2 + 1 = 3, so p2 = 3 since N2 happens to be prime. Continue, unt ...
... example 10 is irrational. 5. in Euclid’s classical proof, a prime outside {p1 , p2 , . . . , pn } is found by considering prime factors of Nn = p 1 p 2 · · · p n + 1 . Do this beginning with {2}, next being {2, p2 }, where in fact N2 = 2 + 1 = 3, so p2 = 3 since N2 happens to be prime. Continue, unt ...
1.7 #6 Meagan
... Because ! is not for ! ! = −2, we can conclude that there does not exist a point on the graph ! ! = ! ! + 6! − 1 where the slope of the tangent line equals zero, and therefore ! ! does not have more than one real solution. This contradiction proves the original statement to be true. ...
... Because ! is not for ! ! = −2, we can conclude that there does not exist a point on the graph ! ! = ! ! + 6! − 1 where the slope of the tangent line equals zero, and therefore ! ! does not have more than one real solution. This contradiction proves the original statement to be true. ...
Proposition: The following properties hold A ∩ B ⊆ A, A ∩ B ⊆ B, A
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
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.