Posts Tagged ‘empty’
Tuesday, January 5th, 2010
How to Determine Dedekind Cuts
Description
A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference.
Overview
A Dedekind cut is a partition of rational numbers into two non-empty sets A and B, such that all elements of A are less than elements of B, and A has no greatest element. The cut itself is a gap that is located between A and B, which is normally found by creating a new, irrational number, and setting it in the gap. What irrational number you use depends on what numbers you have partitioned into the two sets. It is like the number line of advanced algebra, that has both rational and irrational numbers on it instead of just integers. The Dedekind cut was named after Richard Dedekind.
Tags: algebra, between, cut, Dedekind, elements, empty, gap, greater, integer, irrational, less, line, non, non-empty, numbers, partition, rational, Richard, sets, than
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Introduction to Nested Intervals
Description
A detailed tutorial on nested intervals and the nested interval theorem. Step by step tutorial including several examples of nested intervals for reference.
Overview
Nested intervals means to have one interval (or multiple intervals) inside of another interval. The intervals will get smaller and smaller the more you add, until they will finally dimish entirely. There is a theorem for nested intervals, called the nested interval theorem. It states that if A_n = [a_n, b_n] is a sequence of closed intervals such that A_n+1 is a subset of A_n for all n belonging to the set of natural numbers, then the union over A_n is not an empty set.
Tags: algebra, closed, empty, interval, natural, nested, number, open, sequence, set, subset, theorem
Posted in Algebra | No Comments »
Thursday, October 29th, 2009
Overview of Symmetric Relations
Description
A detailed tutorial on the property of symmetric relations. Step by step tutorial including several examples of symmetric relations for reference.
Overview
A symmetric relation can be mathematically defined as for all x, y, and z belonging to A, if x R y and y R z, then x R z. In this statement, A is a set, and R is a relation of that set. An empty set is considered to be symmetric. Since a symmetric relation is defined by a conditional sentence, a proof for the symmetric property of relations would be written as a direct proof.
Tags: conditional, direct, discrete math, empty, equal, equivalence, married, odd, proof, property, r, relation, set, symmetric, x, y
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Overview of Transitive Relations
Description
A detailed tutorial on the property of transitive relations. Step by step tutorial including several examples of transitive relations for reference.
Overview
A transitive relation can be mathematically defined as for all x and y belonging to A, if x R y, then y R x. In this statement, A is a set, and R is a relation of that set. An empty set is considered to be transitive. Since a transitive relation is defined by a conditional sentence, a proof for the transitive property of relations would be written as a direct proof.
Tags: conditional, direct, discrete math, divides, empty, equal, equivalence, great, greater, implies, proof, property, r, relation, set, subset, transitive, x, y, z
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Overview of Reflexive Relations
Description
A detailed tutorial on the property of reflexive relations. Step by step tutorial including several examples of reflexive relations for reference.
Overview
A reflexive relation can be mathematically defined as for all x belonging to A, x R x. In this statement, A is a set, and R is a relation of that set. If the relation is an empty set, then it is not reflexive, unless the set itself happens to be an empty set. When writing a proof for a reflexive relation, you must attempt to prove that (x, x) does not belong to R. If you cannot prove this, then you know that the relation must be reflexive.
Tags: discrete math, divide, empty, equal, equvalence, greater, less, proof, property, r, reflexive, relation, set, subset, x
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Introduction to Equivalence Relations
Description
A detailed tutorial on equivalence relations and how to find them. Step by step tutorial on finding equivalence relations for reference.
Overview
An equivalence relation is a relation that specifies how a set can be split into subsets. Relations can only be considered equivalence relations if they are reflexive, symmetric, or transitive. It is possible for an equivalence relation to be one of these, two of these, or all three of these, If the relation is none of them, then it is not an equivalence relation. An empty set is considered to be an equivalence relation, because it is both symmetric and transitive.
Tags: discrete math, element, empty, equivalence, reflexive, relation, set, subset, symmetric, transitive
Posted in Discrete Math | No Comments »
Thursday, September 24th, 2009
The History of the Number Zero
Description
A detailed tutorial on the history of the number zero. Step by step tutorial including several citations of the history of the number zero for reference.
Overview
Zero is a number we’ve heard about a lot. It’s not a counting number, it’s not negative or positive, it’s not even or odd. It’s not a prime number, it doesn’t even really fit the definitions of a real number or a whole number although it is considered to be both. It is certainly one of the most interesting numbers you can work with. In writing, 0 is distinguished from the capital letter O by either being a bit smaller or having a bit more of an oval shape. Often when handwriting as opposed to typing a line will be drawn through the zero, although this can be confused with an empty set if you are learning set theory. The name zero came from several different lanuages, in which words similar to zero translated to “is empty” “nothing”, and “void”. When doing calculations you must be sure to know the difference between 0 and NaN – “not a number”. Often things that look like they should be zero (0 / 0, for example) are really not numbers at all.
Tags: 0, arithmetic, empty, even, Math, NaN, negative, nil, not a number, nothing, nought, null, number, odd, oh, positive, prime, real, void, whole, zero
Posted in Arithmetic | No Comments »
