1 - UCSD Mathematics
... of the roots of a polynomial. These are now called Galois groups. See Ian Stewart, Why Beauty is Truth, for some of the story of Galois and the history of algebra. Another area that leads to our subject is number theory: the study of the integers. The origins of this subject go back farther than Euc ...
... of the roots of a polynomial. These are now called Galois groups. See Ian Stewart, Why Beauty is Truth, for some of the story of Galois and the history of algebra. Another area that leads to our subject is number theory: the study of the integers. The origins of this subject go back farther than Euc ...
logic for computer science - Institute for Computing and Information
... Gottlob Frege, a German mathematician working in relative obscurity. Frege aimed to derive all of mathematics from logical principles, in other words pure reason, together with some self-evident truths about sets. (Such as 'sets are identical if they have the same members' or 'every property determi ...
... Gottlob Frege, a German mathematician working in relative obscurity. Frege aimed to derive all of mathematics from logical principles, in other words pure reason, together with some self-evident truths about sets. (Such as 'sets are identical if they have the same members' or 'every property determi ...
Primitive sets with large counting functions
... Via a change of variables, we obtain (2). Another question one might consider is what conditions on the distribution of a set A of natural numbers forces A to have a large primitive subset. It is not too difficult to see that if an infinite set A contains no primitive subset of size k, then A(x) k ...
... Via a change of variables, we obtain (2). Another question one might consider is what conditions on the distribution of a set A of natural numbers forces A to have a large primitive subset. It is not too difficult to see that if an infinite set A contains no primitive subset of size k, then A(x) k ...
Coinductive Definitions and Real Numbers
... The Muller-sequence described in the previous chapter shows that there are cases in which floating point arithmetic can not be used. This section briefly describes some of its alternatives, including a slightly more exhaustive account of exact real arithmetic than that given in the introduction. A m ...
... The Muller-sequence described in the previous chapter shows that there are cases in which floating point arithmetic can not be used. This section briefly describes some of its alternatives, including a slightly more exhaustive account of exact real arithmetic than that given in the introduction. A m ...
Bounded negativity of Shimura curves
... Here a lattice is called irreducible if it does not have a finite index subgroup that splits as a product of two lattices. Our geometric definition of Shimura varieties differs from the arithmetic literature on this subject where Shimura varieties are typically not connected. It is the point of view ...
... Here a lattice is called irreducible if it does not have a finite index subgroup that splits as a product of two lattices. Our geometric definition of Shimura varieties differs from the arithmetic literature on this subject where Shimura varieties are typically not connected. It is the point of view ...
EXHAUSTIBLE SETS IN HIGHER-TYPE
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
EXHAUSTIBLE SETS IN HIGHER
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
DISCRETE MATHEMATICAL STRUCTURES - Atria | e
... Power Set: The collection of all subsets of a set A is called the power set of A, and is represented P(A). For instance, if A = {1, 2, 3}, then P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, A} . Multisets: Two ordinary sets are identical if they have the same elements, so for instance, {a, a, b} ...
... Power Set: The collection of all subsets of a set A is called the power set of A, and is represented P(A). For instance, if A = {1, 2, 3}, then P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, A} . Multisets: Two ordinary sets are identical if they have the same elements, so for instance, {a, a, b} ...
Continuous Markovian Logic – From Complete ∗ Luca Cardelli
... In this context we prove the strong robustness theorem: d(P, φ) ≤ d(P, ψ) + d(φ, ψ). In case that d is not computable or it is very expensive, one can use our finite model construction to e ψ) = max{|d(P, φ) − d(P, ψ)|, P ∈ Ωp [φ, ψ]}, where Ωp [φ, ψ] approximate its value. Let d(φ, is the finite mo ...
... In this context we prove the strong robustness theorem: d(P, φ) ≤ d(P, ψ) + d(φ, ψ). In case that d is not computable or it is very expensive, one can use our finite model construction to e ψ) = max{|d(P, φ) − d(P, ψ)|, P ∈ Ωp [φ, ψ]}, where Ωp [φ, ψ] approximate its value. Let d(φ, is the finite mo ...
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.