Posts Tagged ‘discrete math’
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
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Tuesday, October 27th, 2009
The Range of Relations
Description
A detailed tutorial on the range of relations. Step by step tutorial including several examples of the range of relations for reference.
Overview
The range of a relation is denoted as Rng(R) and looks like a normal set. For each ordered pair in a relation, there are two endpoints, x and y. The range is the set of all the y endpoints – that is to say, all the endpoints that come second in the ordered pair. If you are taking the range of the inverse of a relation, then that would be all the x endpoints. When writing the range, the notation used is just the normal notation, not the ordered pair notation.
Tags: cartesian, coordinates, discrete math, element, endpoint, ordered pair, range, relations, second, set, subset
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Tuesday, October 27th, 2009
The Domain of Relations
Description
A detailed tutorial on the domain of relations. Step by step tutorial including several examples of the domain of relations for reference.
Overview
The domain of a relation is denoted as Dom(R) and looks like a normal set. For each ordered pair in a relation, there are two endpoints, x and y. The domain is the set of all x endpoints – that is to say, all the endpoints that come first in the ordered pair. If you are taking the domain of the inverse of a relation, then that would be all the y endpoints. When writing the domain, the notation used is just the normal notation, not the ordered pair notation.
Tags: cartesian, coordinates, discrete math, domain, element, endpoint, First, ordered pair, relations, set, subset
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Tuesday, October 27th, 2009
The Inverse of Relations
Description
A detailed tutorial on the inverse of relations. Step by step tutorial including several examples of the inverse of relations for reference.
Overview
Inverse is a term you should be familiar with. An inverse operation is one that undoes the original operation. But what is an inverse relation? When you take the inverse of a relation, you are switching the endpoints in every ordered pair in the original relation. For each ordered pair in the relation, instead of being written as (x, y) it will now be written as (y, x).
Tags: cartesian, coordinates, discrete math, endpoint, inverse, operation, ordered pair, relations, x, y
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Tuesday, October 27th, 2009
An Introduction to Relations
Description
A detailed tutorial on the introduction to relations. Step by step tutorial including several examples of the introduction to relations for reference.
Overview
A relation is defined as an ordered pair. However, that is not entirely accurate. A relation could either be an ordered pair or a set of ordered pairs. A relation can be used with either one or more normal sets, or one Cartesian product set. When used with a normal set, it is a set of ordered pairs. When used with a Cartesian product, it is the power set of that set.
Tags: cartesian, coordinates, discrete math, element, ordered pair, power, product, relation, set, subset, theory
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Tuesday, October 27th, 2009
Cartesian Products in Set Theory
Description
A detailed tutorial of Cartesian products in set theory. Step by step tutorial including several examples of Cartesian products in set theory for reference.
Overview
A Cartesian product is an operation that can be performed in set theory. It is named not for the multiplication that occurs, but for the way the resulting set is written: it is written in ordered pairs, just like Cartesian coordinates. Two sets are said to be multiplied, such as A and B. Whichever set is written first in the operation has its first coordinate written with the second coordinate of the second set. This continues until all coordinates have been used at least once.
Tags: cartesian, coordinates, discrete math, element, multiplication, operation, ordered pair, product, set, subset, theory
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Thursday, October 22nd, 2009
Inductive Sets in Set Theory
Description
A detailed tutorial on inductive sets in set theory. Step by step tutorial including several examples of inductive sets in set theory for reference.
Overview
An inductive set is a continuous set of natural numbers that follows a basic pattern of n + 1. This means that for all numbers in the set, that number plus the number one must also be included in the set.The set does not need to include all natural numbers – that is, the set may start at any natural number provided it is greater than or equal to one. However, the set must continue to infinity or it cannot be considered an inductive set.
Tags: -1, addition, complete, continuous, discrete math, element, equal, greater, induction, inductive, infinity, mathematical, natural, numbers, one, pattern, principle, set, subset, theory
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Tuesday, October 20th, 2009
Families of Sets in Set Theory
Description
A detailed tutorial on families of sets. Step by step tutorial including several examples of families or collections of sets for reference.
Overview
Families of sets are closely linked with indexed sets – the only sets that can be indexed are families of sets. A family of sets is basically a set of sets. An example would be a power set (the set of all subsets of a set). Unions and intersections can also be performed with families of sets. Instead of concerning just two sets, they concern every single set in the family of sets. The union and intersection over a family of sets are known as extended set operations.
Tags: collections, discrete math, elements, extended, families, index, indexed, intersection, operations, power set, set, set theory, subset, union
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Tuesday, October 20th, 2009
How to Write Proofs by Exhaustion
Description
A detailed tutorial on writing proofs by exhaustion. Step by step tutorial including several examples of how to write proofs by exhaustion for reference.
Overview
A proof by exhaustion is one of the easier types of proofs to write. All this proof involves is testing cases – every case possible for what you are trying to prove. This can be made easier by using variables instead of numbers, or by testing for an even number and odd number, positive and negative number, etc. That way you do not have to test many numbers in order to prove. If even one of the cases does not work out, then whatever you are testing for has been disproven.
Tags: cases, discrete math, disproven, even, exhaustion, Math, method, negative, odd, positive, possibilities, proof, proofs, proven, variable, write
Posted in Discrete Math | No Comments »
Thursday, October 15th, 2009
Complements in Set Theory
Description
A detailed tutorial on complements in set theory. Step by step tutorial including several examples of complements in set theory for reference.
Overview
In set theory, a complement is the opposite of something. It works a little like negation, in that the complement of a set is everything but that set. The way to find this is to subtract the set from its universe, which is a larger set that the set you are taking a complement of belongs to. You can think of your set as a subset of the universe.
Tags: complement, discrete math, elements, Math, negation, opposite, set, set theory, subset, universe
Posted in Discrete Math | No Comments »
