Set Theory/Sets

Sets and elements A mathematical set is defined as an unordered collection of distinct elements. That is, elements of a set can be listed in any order and elements occurring more than once are equivalent to occurring only once. We say that an element is a member of a set. An element of a set can be anything. It's easiest to begin with only numbers as elements. For that reason, most of the examples in this book will only include numbers, but this is only a technique to make the topic less abstract. Terminology For a set having an element , the following are all used synonymously: is a member of is contained in is included in is an element of the set contains includes Notation We specify a set by specifying its members. The curly brace notation is used for this purpose. 1.4em is the set containing 1, 2, 3 as members. Or, {mother, this ipod, my school, the planet Jupiter, 12} is also a set. The curly brace notation can be extended to specify a set by specifying a rule for set membership. ("|" means "such that".) 1.4em is again the set containing 1, 2, and 3 as members. 1.4em is the set of all natural numbers. This form or representing set can be generalized as: 1.4em where is a statement about the variable . The set defined by above notation is a set of all objects such that is true. **** (For a concrete example, consider . Here the property is “” Thus, is the set of all real numbers whose square is one.). EXPLANATION: [A set may be defined by a property. For instance, the set of all planets in the solar system, the set of all even integers, the set of all polynomials with real coefficients, and so on. For a property and an element of a set , we write to indicate that has the property . Then the notation indicates that the set consists of all elements of having the property . The vertical bar | is commonly read as “such that,” and can be also written using a colon instead. So is an alternative notation for . For a concrete example, consider . Here the property is . Thus, is the set of all real numbers ( of (i.e. 1)) whose square is one.] *** A modified epsilon notation is used for set membership. Thus 1.4em means that is a member of . We can also say that is not a member of : 1.4em Characteristics of sets A set is uniquely identified by its members. 1.4em Moreover, the sets are said to be equal if and only if every element of is also an element of , and every element of is an element of . All the above expressions specify the same set even though the concept of an even prime is different from the concept of a positive square root. Repetition of members is inconsequential in specifying a set. The expressions 1.4em all specify the same set. Sets are unordered. The expressions 1.4em all specify the same set. Sets can have other sets as members. There is, for example, the set 1.4em Some special sets As stated above, sets are defined by their members. However some sets are given names to ease referencing them. The set with no members is the empty or null set. The expressions 1.4em all specify the empty set. A set with exactly one member is called a singleton. A set with exactly two members is called a doubleton. Thus is a singleton and is a doubleton. Subsets, power sets, set operations Subsets A set is a subset of set if every member of is a member of . We use the horseshoe notation to indicate subsets. The expression 1.4em says that is a subset of . The empty set is a subset of every set. Every set is a subset of itself. A proper subset of is a subset of that is not identical with . The expression 1.4em says that is a proper subset of . Power sets A power set of a set is the set of all its subsets. is used for the power set. Note that the empty set and the set itself are members of the power set. 1.4em Union The union of two sets A and B, written , is the set that contains all the members of A and all the members of B (and nothing else). That is, 1.4em As an example, 1.4em Intersection The intersection of two sets , written , is the set that contains everything that is a member of both and (and nothing else). That is, 1.4em As an example, 1.4em Two sets are disjoint if their intersection is empty. That is, if and are disjoint sets, 1.4em Relative complement The relative complement of , denoted (sometimes ), is the set containing all the members of that are not members of . That is, 1.4em As an example, 1.4em Absolute complement If we define a universe, or a set containing all of the elements we wish to consider, then we can discuss the absolute complement of a set. For a universe , define the absolute complement of a subset of to be 1.4em The absolute complement of is denoted by (according to the ISO 31-11 standard) if is fixed. Some properties of set operations Union and intersection Based on the preceding definitions, we can derive some useful properties for the operations on sets. The proofs of these properties are left as an exercise to the reader. The union and intersection operations are symmetric. That is, for sets 1.4em Furthermore, they are associative. That is, for sets 1.4em Furthermore, union distributes over intersection and intersection distributes over union. That is, for sets 1.4em De Morgan's laws Two important propositions for sets are De Morgan's laws. They state that, for sets 1.4em When is a universe to which and belong, De Morgan's laws can be stated more simply as, 1.4em Families of sets A set of sets is usually referred to as a family or collection of sets. Often, families of sets are written with either a script or Fraktur font to easily distinguish them from other sets. For a family of sets , define the union and intersection of the family by, 1.4em For a family of sets, we say that it is pairwise disjoint if any two distinct sets we choose from the family are disjoint. pt:Teoria dos conjuntos/Conjuntos