George Boole (1815-1864) England :

 

Boole believed that there was a close analogy between symbols that represent logical interactions and algebraic symbols. He used mathematical symbols to express logical relations. Although computers did not exist in his day, Boole would be pleased to know that his Boolean algebra is the basis of all computer arithmetic. As the inventor of Boolean logic-the basis of modern digital computer logic - Boole is regarded in hindsight as a founder of the field of computer science.

 

Definition :

A set is a collection of well-defined objects. The objects in a set are called elements or members of that set.

 

Introduction :

The concept of set is one of the fundamental concepts in mathematics. The notation and terminology of set theory are useful in every part of mathematics. So, we may say that set theory is the language of mathematics. This subject, which originated from the works of George Boole (1815-1864) and Georg Cantor (1845-1918) in the later part of the 19th century, has had a profound influence on the development of all branches of mathematics in the 20th century. It has helped in unifying many disconnected ideas and thus facilitated the advancement of mathematics. Some operations like union, intersection and difference of two sets. Here, we shall learn some more concepts relating to sets and another important concept in mathematics, namely, function. First, let us recall basic definitions with some examples. We denote all positive integers (natural numbers) by N and all real numbers by R. Here, “well-defined” means that the criteria for deciding if an object belongs to the set or not, should be defined without confusion. For example, the collection of all “tall people” in Chennai does not form a set, because here, the deciding criteria “tall people” is not clearly defined. Hence, this collection does not define a set.

 Notation 

We generally use capital letters like A, B, X, etc. to denote a set. We shall use small letters like x, y, etc. to denote elements of a set. We write x Y ! To mean, x is an element of the set Y. We write t Y b To mean, t is not an element of the set Y.

 Examples

 (i) The set of all high school students in Tamil Nadu.

(ii) The set of all students either in high school or in college in Tamil Nadu.

(iii) The set of all positive even integers.

(iv) The set of all integers whose square is negative.

(v) The set of all people who landed on the moon.

 

Let A B, C D, and E denote the sets defined in (i), (ii), (iii), (iv), and (v) respectively. Note that the square of any integer is an integer that is either zero or positive, and so there is no integer whose square is negative. Thus, set D does not contain any element. Any such set is called an empty set. We denote the empty set by z.

 

 Definition

(i) A set is said to be a finite set if it contains only a finite number of elements in it.

(ii) A set that is not finite is called an infinite set.

 Observe that set A given above is a finite set, whereas set C is an infinite set. Note that the empty set contains no elements in it. That is, the number of elements in an empty set is zero. Thus, the empty set is also a finite set.

 Definition

(i) If a set X is finite, then we define the cardinality of X to be the number of elements in X. the Cardinality of a set X is denoted by n X().

(ii) If a set X is infinite, then we denote the cardinality of X by a symbol 3.

 Now looking at sets A and B, in the above examples, we see that every element of A is also an element of B. In such cases, we say A is a subset of B.

 Subset Let X Y and be two sets. We say X is a subset of Y if every element of X is also an element of Y. That is, X is a subset of Y if z X ! Implies z Y! It is clear that every set is a subset of itself. If X is a subset of Y, then we denote this by X Y 3.

 Set Equality

Two sets X Y and are said to be equal if both contain exactly the same elements. In such a case, we write X Y =. That is, X Y = if and only if X Y 3 and Y X 3.

  Power Set

 Given a set A, let P A() denote the collection of all subsets of A. The set P A() is called the power set of A. If n(A) = m, then the number of elements in P A() is given by n(P(A)) = 2 m. For example, if A = {a, b, c}, then P A() = {z, {ab}, {}, {ca}. {, BA}, {, CB}, {, c}, {, ABC, }} and hence n(P(A)) = 8. Now, given two sets, how can we create new sets using the given sets? One possibility is to put all the elements together from both sets and create a new set. Another possibility is to create a set containing only common elements from both sets. Also, we may create a set having elements from one set that are not in the other set. The following definitions give a precise way of formalizing these ideas. We include a Venn diagram next to each definition to illustrate it.