Closure laws (a) There exists a unique set (b) There exists a unique set. Properties of universe. Properties of null set. Properties of inclusion. Properties of complement. Consistency property The conditions are equivalent.
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Difference of two sets. The set consisting of all elements of a set A that do not belong to a set b short is called the love difference of a and b and denoted by a -. If a a, b, c, d and b b, c, e, f, g then a - b a,. The union, intersection, difference and complement of sets can be depicted graphically by means of Venn diagrams. In a venn diagram the universe u is represented by points within a rectangle and sets a, b, c, etc. Are represented by points inside circles within the rectangle. Figure 1 graphically depicts the union ab of two sets a and b, figure 2 depicts the intersection ab of two sets a and b, fig. 3 depicts the difference a - b of two sets a and b and Fig. 4 depicts the complement of a set. Let a, b, c be any three subsets of a universe. Then the following laws hold:.
The collection of all subsets of the set. The power set of a set a, denoted by p (A) or 2a, is the set consisting of all subsets. The power set of a a, b, c is the set P (a) a, a, b, a, c, listing b, c a, b, c, in general, if a is finite with n elements, p (a) will have 2n elements. The union of two sets a and b is the set consisting of all elements in A plus all elements in b and is denoted by ab or. If a a, b, c, d and b b, c, e, f, g then ab a, b, c, d, e, f,. The intersection of two sets a and b is the set consisting of all elements that occur in both a and B (i.e. All elements common to both) and is denoted by ab, a b. If a a, b, c, d and b b, c, e, f, g then ab b,.
If one of the sets father's is a subset of the other. Two sets a and b are said to be not comparable if a b and e thus note that if two sets a and b are not comparable there is necessarily an element in A that is not in b and an element. Two sets are called disjoint if they have no elements in common. The intersection of the sets is the null set. A system of more than two sets is pairwise disjoint (sometimes called simply disjoint) if every pair of sets in the system is disjoint. Sometimes the members of a set are sets themselves. If thesis the members of a set a are themselves sets it is common to call a a family of sets or a class of sets rather than call it a set of sets. A family of lines in geometry can be regarded as a set of sets since lines can be regarded as sets of points. Power set of a set.
The complement of a set A with respect to a given universal set u is the set of elements in U that are not. The complement of a is typically denoted by Ac or A'. Finite and infinite sets. A finite set is a set with a finite number of elements and an infinite set is one with an infinite number of elements. The set of all black cats in France is a finite set. The set of all even integers is an infinite set. Two sets a and b are said to be comparable if a b or b.
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The null set is considered to be a subset of big every set. Often a discussion involves subsets of some particular set called the universe of discourse (or briefly universe universal set or space. The elements of a space are often called the points of the space. We denote the universal set. The set of all even integers could be considered a subset of a universal set consisting of all the integers.
Or they could be considered a subset of a universal set consisting of all the rational numbers. Or of all the real numbers. Often the universal set may not be explicitly stated and it may be unclear as to just what. At other times it will be clear. Complement of a set.
Any set a, each of whose elements is also an element of s, is said to be contained in s and called a subset. The sets a 5, b 3, 4, 5, and c 6, 7) are all subsets of s 1, 2, 3, 4, 5, 6, 7,. Reviewing the definition we note that the entire set S qualifies as a subset. Thus any set is a subset of itself. We write a is a subset of s as. If a is a subset of s and a s, then a is called a proper subset.
In the above example sets a, b and c are all proper subsets. If a is a subset of s, then we can also write s a which is read s is a superset. Some authors use a s to indicate that a is a subset of s and reserve a b to indicate that a is a proper subset. If a b and b c, then. The empty (or null) set. However illogical it may seem, it is convenient and useful to have the concept of an empty set, a set containing no elements. We call such a set the empty (or null) set and denote.
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Some authors use the colon : instead of the bar. E s x: x is plan an even integer. Sets consisting of the same elements. Two sets a and b are said to be long equal if every element of a is an element of b and every element of b is an element. The equality of sets a and b is denoted by. Inequality of two sets a and b is denoted by. The sets a 3, 4, 5, 6 and b 5, 3, 4, 6 are equal since the order in which set elements are listed is immaterial. The sets a 4, 5, 6 and b 4, 5, 5, 6 are equal since repeating an element of a set does not change the set. Let S be a given set.
such as a, b, x, s, etc. When the roster method is used to define a set, the elements of the set are usually enclosed in braces and separated by commas. S 3, 5, 7, 9 is the set S consisting of elements 3, 5, 7,. S means that x is an element of S; x S means x is not an element. To indicate a set of objects having the property p, the notation x x has the property p is used. The notation x is called a set builder. The bar is read such that. S x x is an even integer, which is the set of all even integers.
Thus the mathematical meaning of the word set summary is the same as the regular, nontechnical meaning of the word. all points in a given line segment - all lines through a given point in space - the set of all rational numbers - all solutions of the equation 3x2 2y all citizens of England - all rivers of Mexico, the individual objects. Synonyms for set: Collection, class, aggregate, ensemble. Methods of defining sets. A set can be defined in either of two ways: 1) explicitly listing each of the elements of the set. The set 2, 3, 4,. 2) describing the set by stating properties that define. The set of all black cats in France.
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Union, intersection, complement, difference. Partially, linearly and well ordered sets. M, website owner: James Miller, home,. Info, mail, set theory. Any statement collection of objects. No restriction is placed on the nature of the objects in a set. They can be anything: points, lines, numbers, people, countries, etc.