Posts Tagged ‘set theory’
Friday, December 18th, 2009
How to Define Cardinal Numbers
Description
A detailed tutorial on the definition of cardinal numbers. Step by step tutorial including several examples of how to define cardinal numbers for reference.
Overview
Cardinal numbers are natural numbers that are used to measure cardinality of sets. Cardinality is a fancy way of saying the size of a set. This means the cardinality is the number of elements in a set, provided that the set is finite. If the set is infinite, something called a transfinite cardinal number is used to describe the cardinality of the set. Cardinal numbers are a very important part of set theory, even though they are not studied often or used constantly.
Tags: abstract, algebra, analysis, cardinal, cardinality, combinatorics, elements, finite, infinite, mathematical, measure, natural, number, set, set theory, size, transfinite
Posted in Algebra | No Comments »
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
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 »
Thursday, October 15th, 2009
Difference in Set Theory
Description
A detailed tutorial of difference in set theory. Step by step tutorial including several examples of difference in set theory for reference.
Overview
Difference is what you get after subtracting two numbers – or two sets. As with other examples of subtraction, order is very important for difference in set theory. Unless two sets are identical, you will end up with a different answer depending on the order. Difference is very often used in conjunction with union and intersection of sets or power sets.
Tags: difference, discrete math, element, empty set, intersection, Math, number, order, power set, set, set theory, subset, subtract, subtraction, union
Posted in Discrete Math | No Comments »
Tuesday, October 13th, 2009
Empty Set in Set Theory
Description
A detailed tutorial on the empty set. Step by step tutorial including several examples and a description of the properties of the empty set for reference.
Overview
The empty set is a unique set in set theory that means a set composed of nothing. In an empty set, there are no elements at all. The empty set has one very unique property – it is the subset of all sets. The set of all natural numbers up to infinity? It’s a subset. The set of prime numbers less than 20? It’s a subset of that, too. It is also a subset of itself – although that is not particurarly unique. The empty set is not used in equations, but can be used to define them.
Tags: difference, discrete math, element, empty set, intersection, Math, none, set, set theory, subset, union, unique, zero
Posted in Discrete Math | No Comments »
Tuesday, October 13th, 2009
Power Sets in Set Theory
Description
A detailed tutorial on power sets. Step by step tutorial including several examples of power sets and how to perform operations of power sets for reference.
Overview
Power sets are defined as a set of all subsets. So for example, say you have a set A. The power set of A would be the set of all possible subsets of A. Power sets can also be used in normal operations, such as intersections and unions. All you do is find all possible subsets of both sets you are working with, and solve the problem like you would with a normal set.
Tags: difference, discrete math, element, empty set, intersection, Math, power, set, set theory, subset, union
Posted in Discrete Math | No Comments »
Thursday, October 8th, 2009
Subsets in Set Theory
Description
A detailed tutorial on how to identify subsets of a set. Step by step tutorial including several examples of how to find subsets in a set for reference.
Overview
Each set in set theory has a certain amount of subsets. There is an easy way figure out how many subsets a set has. Pretend that every element of a set is 2, and multiply them together. This will be your number of subsets. For example, if you have three elements, you will have 8 subsets, because 2 cubed (which is 2 to the power of 3, or 2 times 2 times 2) is equal to 8. Now that you have determined how many subsets there are, you have to figure out what they are. A subset is defined as any set containing all or part of a set. Two subsets are going to be the set itself, and an empty set. Sometimes they are your only subsets. Now, following the definition, a subset must be all possible sets. This means, sets of one element - one for each element in your set. In addition to that, you may have sets of two elements – one for each possible combination of elements in your set. This should be continued until you have reached the maximum number of elements in the set you atarted out with.
Tags: combination, discrete math, element, empty set, exponent, Math, multiplication, number, set, set theory, subset, to the power, value
Posted in Discrete Math | No Comments »
Friday, October 2nd, 2009
Disjoint Sets in Set Theory
Description
A detailed tutorial on disjoint sets. Step by step tutorial including several examples of disjoint sets and how to identify disjoint sets for reference.
Overview
A disjoint set is a term applied in set theory when two or more sets have no elements in common. For example, the sets {1, 2, 3} and {7, 8, 9} are disjoint sets because none of the numbers in the sets are the same. The formal way to say this is that two sets are disjoint sets if their intersection creates an empty set, in other words, nothing at all. An intersection is when you only take the values that are found in both sets. If none of the values are the same, this would be an empty set. Disjoint sets can be classified into further categories of piecewise, pairwise, or mutually disjoint provided that in a collection, at least two sets are disjoint.
Tags: collection, discrete math, disjoint, elements, empty set, intersection, Math, mutually, pairwise, piecewise, set theory, sets, value
Posted in Discrete Math | No Comments »
Friday, October 2nd, 2009
The Union and Intersection of Sets
Description
A detailed tutorial on the union and intersection of sets. Step by step tutorial including several examples of the union and intersection of sets for reference.
Overview
Set theory is a branch of mathematics that deals with sets of numbers and the way that they interact with each other. One part of set theory is union and intersection. Union is represented by the symbol U, and means to combine the numbers in a set. The union of A and B states that for all x in A and B, the union contains all x in A and all x in B. Intersection is represented by an upside-down letter U, and means to only use numbers that are found in both sets. The intersection of A and B states that for all x in A and B, the intersection contains all x found in both A and B. The definitions might seem similar, but they are different.
Union:
A = {1, 2, 3, 4}, B = {2, 3, 6, 7}. The union would be {1, 2, 3, 4, 6, 7}.
Intersection:
A = {1, 2, 3, 4}, B = {2, 3, 6, 7}. The intersection would be {2, 3}
Tags: a, and, b, belonging to, combine, difference, discrete math, interact, intersection, Math, or, set theory, sets, union, x
Posted in Discrete Math | No Comments »
Tuesday, September 29th, 2009
The Use of Venn Diagrams in Set Theory
Description
A detailed tutorial on using Venn diagrams in set theory. Step by step tutorial including several examples of how to use a Venn diagram in set theory for reference.
Overview
A Venn diagram is something we’ve all heard about before – probably in an english class. Venn diagrams are used as a visual representation for sets of different items, called elements. But it stands to reason that if you can use a venn diagram for sets, you can use them for set theory – a branch of mathematics that uses different sets of numbers. For set theory, they are used to display intersection and unions, and you can fill them out just like you would with a normal Venn diagram.
Tags: algebra, De Morgan's Rules, De Morgan's Theorem, diagram, elements, intersection, set theory, sets, union, Venn
Posted in Algebra | No Comments »
