Powerpoint Source - Mathematics
... • It is a type of fractal and was discovered in 1980 by Benoit Mandlebrot. • It uses Z^2 + C to test complex numbers on the Argand plane to see if they are contained within the boundaries of the set. This set being the region on the Argand plane for which upon repeating this sequence it remains boun ...
... • It is a type of fractal and was discovered in 1980 by Benoit Mandlebrot. • It uses Z^2 + C to test complex numbers on the Argand plane to see if they are contained within the boundaries of the set. This set being the region on the Argand plane for which upon repeating this sequence it remains boun ...
2.! You are given 3 prime numbers a 1, a2, and an, and m, with m
... 2.! You are given 3 prime numbers a1, a2, and an, and m, with m equaling the least common prime of the three prime numbers. Add 1 to m and prove that whether or not m + 1 is prime or not, the number of prime numbers infinite. ...
... 2.! You are given 3 prime numbers a1, a2, and an, and m, with m equaling the least common prime of the three prime numbers. Add 1 to m and prove that whether or not m + 1 is prime or not, the number of prime numbers infinite. ...
Computability
... – Turing recognizable because U recognizes it. – May not halt because U [only] simulates M and M may not halt. • But maybe another technique could be better than M... • This is the halting problem ...
... – Turing recognizable because U recognizes it. – May not halt because U [only] simulates M and M may not halt. • But maybe another technique could be better than M... • This is the halting problem ...
Set theory
... A set is infinite if and only if it has the same size as some proper subset Two arbitrary sets M and N (finite or infinite) are said to be of equal size or cardinality, if and only if there exists a bijection from M onto N. Set M countable if it can be put in one-to-one correspondence with N ...
... A set is infinite if and only if it has the same size as some proper subset Two arbitrary sets M and N (finite or infinite) are said to be of equal size or cardinality, if and only if there exists a bijection from M onto N. Set M countable if it can be put in one-to-one correspondence with N ...
A group is a non-empty set G equipped with a binary operation * that
... Definition: An element a in a group is said to have finite order if a k = e for some positive integer k. In this case, the order of the element a is the smallest positive integer n for which a n = e . The order of a is denoted by o(a) or |a|. An element a is said to have infinite order if a k ≠ e fo ...
... Definition: An element a in a group is said to have finite order if a k = e for some positive integer k. In this case, the order of the element a is the smallest positive integer n for which a n = e . The order of a is denoted by o(a) or |a|. An element a is said to have infinite order if a k ≠ e fo ...
ONTOLOGY OF MIRROR SYMMETRY IN LOGIC AND SET THEORY
... CANTOR'S CONTINUUM HYPOTHESIS: C = 1, where С = 20 > 0 . K.Goedel (1939) and P.Cohen (1962) proved the independence of the CH within the framework of the axiomatic set theory of Zermelo-Fraenkel, but the proof of the CHindependence and the CH-solution are obviously quite different things. The sit ...
... CANTOR'S CONTINUUM HYPOTHESIS: C = 1, where С = 20 > 0 . K.Goedel (1939) and P.Cohen (1962) proved the independence of the CH within the framework of the axiomatic set theory of Zermelo-Fraenkel, but the proof of the CHindependence and the CH-solution are obviously quite different things. The sit ...
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... plane, f (A) + f (B) + f (C) + f (D) = 0. Does it follow that f (P ) = 0 for all points P in the plane? A4 Let S be a set of rational numbers such that (a) 0 ∈ S; (b) If x ∈ S then x + 1 ∈ S and x − 1 ∈ S; and (c) If x ∈ S and x 6∈ {0, 1}, then 1/(x(x − 1)) ∈ S. Must S contain all rational numbers? ...
... plane, f (A) + f (B) + f (C) + f (D) = 0. Does it follow that f (P ) = 0 for all points P in the plane? A4 Let S be a set of rational numbers such that (a) 0 ∈ S; (b) If x ∈ S then x + 1 ∈ S and x − 1 ∈ S; and (c) If x ∈ S and x 6∈ {0, 1}, then 1/(x(x − 1)) ∈ S. Must S contain all rational numbers? ...
Module 5
... Infinite # of elements: {an | n is even}, {an | n is prime}, {an | n is a perfect square} ...
... Infinite # of elements: {an | n is even}, {an | n is prime}, {an | n is a perfect square} ...
Document
... Definition: Two sets A and B have the same cardinality if and only if there exists a bijection (or a one-to-one correspondence) between them, A ~ B. We split infinite sets into two groups: 1. Sets with the same cardinality as the set of natural numbers 2. Sets with different cardinality as the set o ...
... Definition: Two sets A and B have the same cardinality if and only if there exists a bijection (or a one-to-one correspondence) between them, A ~ B. We split infinite sets into two groups: 1. Sets with the same cardinality as the set of natural numbers 2. Sets with different cardinality as the set o ...
Infinity
Infinity (symbol: ∞) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics.In mathematics, ""infinity"" is often treated as if it were a number (i.e., it counts or measures things: ""an infinite number of terms"") but it is not the same sort of number as natural or real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number; see 1/∞.Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.