Normal numbers without measure theory - Research Online
... expansion. A similar definition to the above may be made for a number to be simply normal to other bases. The following result was proved by Émile Borel in 1904. Borel’s Theorem. There is a subset Z of [0, 1) which has measure zero and is such that every number in [0, 1) which is not in Z is simply ...
... expansion. A similar definition to the above may be made for a number to be simply normal to other bases. The following result was proved by Émile Borel in 1904. Borel’s Theorem. There is a subset Z of [0, 1) which has measure zero and is such that every number in [0, 1) which is not in Z is simply ...
Solutions
... whereas the point of this exercise is to go in the other direction, namely, deduce this fact from the existence of primitive roots.) Proof. Let a be a primitive root. Every element of Zpn is of the form ak for some k such that 0 ≤ k < ϕ(pn ). This element has order d = ϕ(pn )/(ϕ(pn ), k) which is th ...
... whereas the point of this exercise is to go in the other direction, namely, deduce this fact from the existence of primitive roots.) Proof. Let a be a primitive root. Every element of Zpn is of the form ak for some k such that 0 ≤ k < ϕ(pn ). This element has order d = ϕ(pn )/(ϕ(pn ), k) which is th ...
Midterm: model solutions and mark scheme File
... by laws of exponents. This equals the right hand side of P(k + 1), so we have proved that P(k) implies P(k + 1) for any integer k ≥ 1. Conclusion: Therefore by induction, P(n) is true for all integers n ≥ 1. Common errors 1A Replacing multiplication by addition. The sign ∏ means that successive term ...
... by laws of exponents. This equals the right hand side of P(k + 1), so we have proved that P(k) implies P(k + 1) for any integer k ≥ 1. Conclusion: Therefore by induction, P(n) is true for all integers n ≥ 1. Common errors 1A Replacing multiplication by addition. The sign ∏ means that successive term ...
ON HIERARCHIES AND SYSTEMS OF NOTATIONS
... Introduction. By a "system of notations" we understand a (1-1)mapping, M, from a set L (a segment of Cantor's second number class) onto a family, F, of disjoint nonempty sets. The members of F may be, for example, sets of expressions, e.g. such expressions as w, uX2, w2, €o, etc. Without loss of gen ...
... Introduction. By a "system of notations" we understand a (1-1)mapping, M, from a set L (a segment of Cantor's second number class) onto a family, F, of disjoint nonempty sets. The members of F may be, for example, sets of expressions, e.g. such expressions as w, uX2, w2, €o, etc. Without loss of gen ...
Lecture 29
... • Our construction insures that all strings derived from S in L(G2) are also in L(G1) ...
... • Our construction insures that all strings derived from S in L(G2) are also in L(G1) ...
Full text
... subject to the initial conditions A(l, 0) = 1, A(l, 1) ='2, with 4(n, /c) = 0 for & < 0 or k > n. This array has been called a Lucas triangle by Feinberg [1], because rising diagonals sum to give the Lucas numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ..., in contrast to the rising diagonal ...
... subject to the initial conditions A(l, 0) = 1, A(l, 1) ='2, with 4(n, /c) = 0 for & < 0 or k > n. This array has been called a Lucas triangle by Feinberg [1], because rising diagonals sum to give the Lucas numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ..., in contrast to the rising diagonal ...
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.