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| Basic Definitions |
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| A well-defined collection of distinct objects is called a set. |
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| Capital letters are usually used to denote or represent a set. |
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| viz., A, B, C, D, . . . . |
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| The elements of sets are usually represented by lower case letters. |
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| viz., a, b, c, x, y, . . . |
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| Study the following examples |
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| 1. The numbers 2, 3, 5 and 8. |
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| 2. The solutions of the equation x2
- 2x - 15 = 0. |
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| 3. The letters of the word "Chase". |
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| 4. People living in our locality. |
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| 5. The students
Mary, Hillary, Catherine. |
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| 6. The workers who are absent from the factory. |
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| 7. The capital cities of Asia. |
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| 8. The positive even integers or the numbers 2, 4, 6, 8, . . . |
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| 9. The rivers in US. |
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| Observe that the sets in the odd numbered examples are defined, that is they are presented by listing its elements and the even numbered examples are defined by stating properties i.e. rules which decide whether the particular element is a member or not. |
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| There are two methods of representing a set. |
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| (i) Roster Method |
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| (ii) Set builder form |
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| (i) Roster Method |
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| This method is also known as tabular method. |
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| In this method, a set is represented by listing all the elements of the set, the elements being separated by commas and are enclosed within flower brackets { }. |
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| Example: |
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| A set containing 2,3,5,7,8 is Written as {2,3,5,7,8} in Roster method. |
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| (ii) Set builder Method |
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| This method is also known as Property method. |
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| In this method, a set is represented by stating all the properties which are satisfied by the elements of the set and not by any other element. |
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| If A contains all values of 'x' for which the condition P(x) is true, then we write A = {x: P(x)}. |
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| Example: |
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| If A is a set of all positive factors of 36, then in Set builder method it is written as |
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| A = {x : x is a factor of 36, x
Î N} |
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| A set is finite if it contains a specific number of elements. |
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| Otherwise, a set is an infinite set. |
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| Examples : |
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| The following are examples of a finite set. |
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- Set of stars in the universe.
- {x : x Î N x < 1010}
- {x : x Î I, x > 0, x is a factor of 156250}
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| Example 2: |
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| The following are examples of an infinite set. |
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- Set of all points on a line segment of 2cm.
- Set of all similar triangles in a plane.
- Set of rational numbers between the integers 1 and 2
- {x : x Î I and x > 2}
- {x: x is an even number}
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| Note: |
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| Example: |
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| E = {x : x2 = 9 and 3x = 6} is
an empty set because there is no number that can satisfy both x2
= 9 and 3x = 6. |
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| 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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| Example: |
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| A = {0} |
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| n (A) = 1 |
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| 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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| 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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| Example: |
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| A = {3, 4, 5, 6}, B = {6, 4, 3, 5} |
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| A = B |
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| Elements in both sets need not be of same order. |
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| The number of elements in a finite set A, is the cardinality of A and is denoted by n(A). |
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| Example: |
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| If A = {0, 7, 1, 2, 4, 11}, then n (A) = 6. |
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| 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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| Usually we denote the universal set by the letter U. |
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| Example: |
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| In plane geometry, the universal set consists of all points in a plane. |
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| 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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| or |
| A set A is said to be a proper subset of B if B contains at least one element, which is not in A. |
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| Example: |
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| 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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| If a set S is finite, say S has n elements, then the power set of S can be shown to have 2n elements. |
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