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Introduction |
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In Mathematics, a well-defined collection of definite objects is called a set. |
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George Cantor is regarded as the "Father of Set theory". |
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The concept of "Sets" is basic in all branches of mathematics. |
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Basic Definitions |
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Set: A well-defined collection of distinct objects is called a set. |
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Notation of Sets: Capital letters are usually used to denote or represent a set. |
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Representation of Sets: There are two methods of representing a set.
(i) Roster Method
(ii) Set builder form. |
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Finite and Infinite Sets: A set is finite if it contains a specific number of elements.
Otherwise, a set is an infinite set. |
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Null Set or Empty Set or Void Set: A set with no elements is an empty set. |
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Singleton Set or Singlets: A set consisting of a single element is called a singleton set or singlet. The cardinality of the singleton set is 1. |
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Equivalent Sets: Two finite sets A and B are said to be equivalent sets if cardinality of both sets are equal i.e. n (A) = n (B). |
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Equal Sets: Two sets A and B are said to be equal if and only if they contain the same elements i.e. if every element of A is in B and every element of B is in A. We denote the equality by A = B. |
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Cardinality of a Set A: The number of elements in a finite set A, is the cardinality of A and is denoted by n(A). |
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Universal Set: In any application of the theory of sets, the members of all sets under consideration usually belong to some fixed large set called the universal set. |
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Subsets: If A and B are sets such that each
element of A is an element of B, then we say that A is a subset of B and
write A Í B. |
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Power Set: The family of all subsets of any set S is called the power set of S. We denote the power set of S by P (S). |
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Some Results of Subsets |
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Prove that Null Set is the subset of every set |
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Every set A is a subset of universal set
U since, by definition all elements of A belong to U. Also the null set
f Í A. |
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The null set f
is a subset of every set, in particular f Í A. By
hypothesis, A Í f. The two conditions imply A =
f. |
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The total number of all possible subsets of a given set containing n elements is 2n |
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Operations on Sets |
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The Operations on Sets are: Union of sets, Intersection of sets, Disjoint sets, Difference of two sets (Relative complement), Symmetric Difference of two sets, Complement of a set. |
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Venn Diagrams [Euler-Venn Diagrams] |
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A Venn diagram is a pictorial representation of sets by set of points in the plane. The universal set U is represented pictorially by interior of a rectangle and the other sets are represented by closed figures viz circles or ellipses or small rectangles or some curved figures lying within the rectangle.
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Algebraic Properties of set operations |
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The Algebraic Properties of set operations are: Idempotent laws, Identity laws, Commutative laws, Associative laws, Distributive laws, De Morgan's Laws. |
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Application of sets |
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The theory of finite sets is, arguably, a definition of Combinatorics. In particular, certain combinatorial topics (e.g. Ramsey theory) have important direct analogues in "combinatorial" set theory.
Since Axiomatic Set Theory is often used to construct the natural numbers (satisfying the Peano axioms, say) it is possible to translate statements about Number Theory to Set Theory. Indeed, a fairly robust theory of arithmetic has been developed for ordinals and cardinals.
Fuzzy sets, or more precisely logic, are applied to topics in Information Theory and Artificial Intelligence since this gives a language in which to discuss decision-making. |
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Summary |
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In roster method of representing a set, all the elements are listed in the set. |
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Two sets are said to be Equivalent sets if the elements of one set can be put in one-one correspondence with the elements of the other set. |
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Two sets are said to be Equal sets if every element of one set is in the other set and vice-versa.
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