- 1. JSS MAHAVIDYAPEETHA MYSORE 04 JSS INSTITUTE OF EDUCATION, SAKALESHPUR Topic- SETS Submitted by Rohith V 1st Year B Ed 2nd semester
- 2. SETS
- 3. SETS A set is a well-defined collection of objects. Here well- defined means it must be particular with reference to all The following points may be noted in writing sets: (i) Objects, elements and members of a set are synonymous terms. (ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc. (iii) The elements of a set are represented by small letters a, b, c, x, y, For example : A={ s,a,k,l,e,h,p,u,r} Here A is set and s,a,k,l,e,p,u,r are element
- 4. REPRESENTATION OF SET There are two methods of representing a set : (i) Roster or tabular form (ii) Set-builder form. Roster or tabular form In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces ● in roaster form, the elements are distinct Example Multiple of 3 between 1 and31 is {3,6,9,12,15,18,21,24,27,30} The word ‘college’ written in roaster form as {c,o,l,e,g}
- 5. Set-builder form All the elements of a set possess a single common property Which is not possessed by any element outside the set. Example: V={x : x=vowels in English alphabet} R={ y: y=colors in rainbow} Roster form Set builder form O={1,3,5,7,9} O={x : x=odd number below 10}
- 6. Empty set: A set which does not contain any element is called the empty set or the void set or null set and is denoted by { } or φ. Finite and infinite set: A set which consists of a finite number of elements is called a finite set otherwise, the set is called an infinite set. Example Finite set => A={1,2,3,4,5,6,7,8,9} Infinite set=> S={Number of stars in sky} Subset: A set A is said to be a subset of set B if every element of A is also an element of B. In symbols, A ⊂ B if a ∈ A ⇒ a ∈ B. Example A={1,2,3,4,5,6} and B={2,3,6,} A ⊂ B
- 7. Equal set: Given two sets A and B, if every elements of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal. If A={2,4,6,8} and B={2,4,6,8} Then set Aand set B are equal set Intervals as subsets of R Let a, b ∈ R and a < b. Then (a) An open interval denoted by (a, b) is the set of real numbers {x : a < x < b} this imples all elements between a & b expect a, b (b) A closed interval denoted by [a, b] is the set of real numbers {x : a ≤ x ≤ b) Means all elements between a &b and a,b (c) Intervals closed at one end and open at the other are given by [a, b) = {x : a ≤ x < b} (a, b] = {x : a < x ≤ b}
- 8. Power set: The collection of all subsets of a set A is called the power set of A. • it is denoted by P(A). • If the number of elements in A = n , i.e., n(A) = n, then thenumber of elements in P(A) = 2n Universal set : • This is a basic set. • in a particular context whose elements and subsets are relevant to that particular context. • It is denoted by English aphabet letter U Example , for the set of vowels in the English alphabet, the universal set can be the set of all alphabets in English. Universal
- 9. Venn Diagrams Venn Diagrams are the diagrams which represent the relationship between sets.
- 10. Union of Sets : The union of any two given sets A and B is the set C which consists of all those elements which are either in A or in B. In symbols, we write C = A ∪ B = {x | x ∈A or x ∈B} Example 1. A={ 1,2,3,4} B={5,6,7,8} C= A ∪ B = {1,2,3,4,5,6 7,8} 2. D={2,3,6,7} E={1,3,7,8,9} F = D ∪ E = {2,3,6,7,1,8,9} or {1,2,3,6,7,8,9} Some properties of the operation of union. (i) A ∪ B = B ∪ A (ii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (iii) A ∪ φ = A (iv) A ∪ A = A (v) U ∪ A = U
- 11. Intersection of sets: The intersection of two sets A and B is the set which consists of all those elements which belong to both A and B. ■ Intersection of set is denoted by ‘∩’ ■ Symbolically, A ∩ B = {x : x ∈ A and x ∈ B} ■ When A ∩ B = φ, then A and B are called disjoint sets. Example 1. A={1,4,5,9} B={1,9,4,7} A ∩ B ={1,4,9} 2. C={2,4,7} D={1,3,5} C ∩ D = φ Some properties of the operation of intersection (i) A ∩ B = B ∩ A (ii) (A ∩ B) ∩ C = A ∩ (B ∩ C) (iii) φ ∩ A = φ ; U ∩ A = A (iv) A ∩ A = A (v) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (vi) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)