Section 2.5
... An important tool the mathematicians use to compare the size of sets is called a one-to-one correspondence. This concept is a way of saying two sets are the same size without counting the numbers in them. We call two sets equivalent if they have the same number of elements. Equivalent sets can be pu ...
... An important tool the mathematicians use to compare the size of sets is called a one-to-one correspondence. This concept is a way of saying two sets are the same size without counting the numbers in them. We call two sets equivalent if they have the same number of elements. Equivalent sets can be pu ...
Mathematicians
... word which means 'witch'. The equation for this bell-shaped curve was given the name 'witch of Agnesi' and it stuck and can be found in some textbooks today. 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science. First woman to be appointed as professor ...
... word which means 'witch'. The equation for this bell-shaped curve was given the name 'witch of Agnesi' and it stuck and can be found in some textbooks today. 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science. First woman to be appointed as professor ...
MAT 140 Discrete Mathematics I
... 1. All the white objects are squares. then it is a square. 2. All the square objects are white. 3. No square objects are white. 4. There is a white object that is larger than every gray object. 5. Every gray object has a black object next to it. 6. There is a black object that has all the gray objec ...
... 1. All the white objects are squares. then it is a square. 2. All the square objects are white. 3. No square objects are white. 4. There is a white object that is larger than every gray object. 5. Every gray object has a black object next to it. 6. There is a black object that has all the gray objec ...
Constructive Mathematics, in Theory and Programming Practice
... Indeed, it is ironic that, having first become interested in constructivism through the persuasive writings of Bishop, in which, as with Brouwer, the use of what became identified as intuitionistic logic was derived from an analysis of his perception of meaningful mathematical practice, we have been ...
... Indeed, it is ironic that, having first become interested in constructivism through the persuasive writings of Bishop, in which, as with Brouwer, the use of what became identified as intuitionistic logic was derived from an analysis of his perception of meaningful mathematical practice, we have been ...
printable
... • Prove that there are some problems that cannot be solved • Show that there are some problems that (are believed to) require an exponential amount of time to solve (NPComplete) • Examine some strategies for dealing with these problems • Along the way, learn how to model computation mathematically, ...
... • Prove that there are some problems that cannot be solved • Show that there are some problems that (are believed to) require an exponential amount of time to solve (NPComplete) • Examine some strategies for dealing with these problems • Along the way, learn how to model computation mathematically, ...
Internal Inconsistency and the Reform of Naïve Set Comprehension
... excluded from RSC comprehension by the Russell exclusion. It is contended that all set descriptions implicated in the known logical antinomies can be shown intuitively to “contain” their own contradictions since the contradiction is inferred by the set description of an invalid description alone; su ...
... excluded from RSC comprehension by the Russell exclusion. It is contended that all set descriptions implicated in the known logical antinomies can be shown intuitively to “contain” their own contradictions since the contradiction is inferred by the set description of an invalid description alone; su ...
Chapter 1: Sets, Operations and Algebraic Language
... not an element of the set of all numbers greater than 1. The closed dot on the second number line indicates that 1 is an element of the set of all number greater than or equal to 1. Whether the dot is open or closed signifies whether the starting number (or ending number) is “in” or “not in” the set ...
... not an element of the set of all numbers greater than 1. The closed dot on the second number line indicates that 1 is an element of the set of all number greater than or equal to 1. Whether the dot is open or closed signifies whether the starting number (or ending number) is “in” or “not in” the set ...
2015Khan-What is Math-anOverview-IJMCS-2015
... thought about the nature of Mathematics: one is the Platonist or Realist (deriving from Plato) and the other is Formalist. The Platonists believe that mathematical objects exist independent of us and inhabit a world of their own. They are not invented by us but rather discovered. Formalists on the o ...
... thought about the nature of Mathematics: one is the Platonist or Realist (deriving from Plato) and the other is Formalist. The Platonists believe that mathematical objects exist independent of us and inhabit a world of their own. They are not invented by us but rather discovered. Formalists on the o ...
Logic and Categories As Tools For Building Theories
... It allows us simultaneously to be both properly specific and general: specific, in that statements about mathematical structures are not really precise until we have specified which structures we are dealing with, and which morphisms we are considering — i.e. which category we are working in. At the ...
... It allows us simultaneously to be both properly specific and general: specific, in that statements about mathematical structures are not really precise until we have specified which structures we are dealing with, and which morphisms we are considering — i.e. which category we are working in. At the ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.