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Notes On Sets

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Published in: Mathematics
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Notes on Mathematics Sets Covering Sets, Types of Sets, Operation in Sets and theorems.

Rizwan B / Chennai

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Qualification: BE

Teaches: Botany, Chemistry, Mathematics, Physics, Biology

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  1. SETS 1. A set is a well-defined collection of objects. 2. The objects of a set are called its members or elements. 3. Elements of a set are listed only once. Notation l. 2. 3. 4. 5. 6. A set is usually denoted by capital letters of the English Alphabets The elements of a set are denoted by small letters of the English alphabets The elements of a set is written within curly brackets } ' If x is an element of a set A or x belongs to A, we write x e A. If x is not an element of a set A or x does not belongs to A, we write x e A. The order of listing the elements of the set does not change the set. Representation of a Set A set can be represented in any one of the following three ways or forms: 1. Descriptive Form. 2. Set-Builder Form or Rule Form. 3. Roster Form or Tabular Form. I.Descriptive Form In descriptive form, a set is described in words. Eg) The set of all vowels in English alphabets. 2.Set Builder Form or Rule Form In set builder form, all the elements are described by a rule. The symbol ':' or stands for "such that". A = {x : x is a vowel in English alphabets}. 3.Roster Form or Tabular Form A set can be described by listing all the elements of the set. Eg) A = {a, e, i, o, u} Three dots ( ... ) is called ellipsis. It represents continuation of elements in that set 1
  2. Cardinality of a set - n(A) The no of Elements in a set is known as cardinality of a set A = { x : x is a number in the first 15 positive multiples of 5} C = { x : x is a planet in solar system} Set types I.Empty Set or Null Set n(A) = 15 A set consisting of no element is called the empty set or null set or void set. It is denoted by Ø or { }. Eg) A={x : x is an odd integer and divisible by 2} 2.Singleton Set A set which has only one element is called a singleton set. Eg) A = {x : 3 < x < 5, x e N} A = {4} 3.Finite Set A set with finite number of elements is called a finite set. Eg) A = {x : x is the planet in the solar system} 4.1nfinite Set 5.Equivalent Sets B is The set of all points on a line. Two finite sets A and B are said to be equivalent if they contain the same number of elements. It is written as A z B. If A and B are equivalent sets, then n(A) = n(B) Eg) A = { Ball, Bat} B = {history, geography}. Here A is equivalent to B because n(A) = n(B) = 2. 2
  3. 6.Equal Sets Two sets are said to be equal if they contain exactly the same elements, otherwise they are said to be unequal. Denoted by A = B In other words, two sets A and B are said to be equal, if (i) every element of A is also an element of B (ii) every element of B is also an element of A Eg) A = {1, 2, 3, 4} and B = {4, 2, 3, 1} Since A and B contain exactly the same elements, A and B are equal sets 7.Subset Let A and B be two sets. If every element of A is also an element of B, then A is called a subset of B. Denoted as A C B. B = {x:x is a set of natural numbers} { -1,-4} Then, Note 1. If A C B and B C A, then A-B. 2. Empty set is a subset of every set. 4. Every set is a subset of itself 8.Proper Subset Let A and B be two sets.lf A is a subset of B and A#B, then A is called a proper subset of B and we write A I B. For example, If A={ 1,2,5} and B={ 1,2,3,4,5} then A is a proper subset of B ie. A cB. 3
  4. 9.Power Set The set of all subsets of a set A is called the power set of 'A' , denoted by P(A). The subsets of A are Ø , {2},{3},{2,3}. The power set of A, P(A) = {Ø ,{2},{3},{2,3}} Note 2. Null set is always a subset of any set 3. If n(A) = m, then = 2m 4. The number of proper subsets of a set A is n[P(A)]—1 = 2 m—l. 10.Universal Set A Universal set is a set which contains all the elements of all the sets under consideration .Denoted by U. Eg) U is the set of all Natural numbers. U={x : x e N}. If A={earth, mars, jupiter}, then U is the planets of solar system. Set Operations 1.Complement of a set — A' , A U — A The Complement of a set A is the set of all elements of U (the universal set) that are not in A. ' A'— {x : xeU, xCA} Eg) U = { x : x is a multiple of 2 between 0 and 20 } Then , A' = A' A' (shaded region) A = { 3,6, 9,12} 4
  5. 2.Union of two Sets U Set of all elements which are either in A or in B or in both. AUB = {x : x GA or xeB} B = { a,b, c,d,e} Sets A and B have common elements Sets A and B are disjoint Properties 3. AUU= U where A is any subset of universal set U 4. ACAUB and BC AUB 5. AUB = BOA (union of two sets is commutative) 3.1ntersection of two sets The intersection of two sets A and B is the set of all elements common to both A and B. AQB and read as A intersection B. : xeA and xeB} 5
  6. Properties 2. AnU = A where A is any subset of universal set U 4.Difference of two sets 3. AnB C A and AnB C B 4. AnB = BnA (Intersection of two sets is commutative) Set of all elements which are in A, but not in B. Denoted by A—B or read as A difference B. A — B means elements in A which are not in B A—B = { x : x e A and x e B} = { y: y e B and y e A}. 7,8, 10} Properties 2. A-B = AnB' 5.Symmetric Difference Don't have common elements in them at the intersection AAB AAB={ x : x e A—B or x e B—A} Eg) A = { x . x is a multiple of 2 between 0 and 10 } B = { x : x is a multiple of 3 between 0 and 10 } A -B = {2, 4, 8,101 A = { ,10} = { 3,6,9 } B 6
  7. Properties I. AAA 2. AAB 3. AAB 4. AAB BAA {x : xeAUB and xCAß)B (AUB) - (AnB) 6. Disioint Set Two sets do not have common elements. 7. Overlapping Sets Two sets have I common element. AnB=Ø A and B are disjoint set A and B are overlapping set 7
  8. l. 2. 3. 1. Note When B CA, the union and intersection of A and B are represented in Venn diagram as shaded region is shaded region is For any two sets A and B, AUB = AQB. A — B Let n(A) = p and n(B) = q then (a) (b) (c) (d) Minimum of n(AUB) = max{p, q} Maximum of n(AUB) = p + q Minimum of = 0 Maximum of n(AQB) = min{p, q} Properties of Set Operations Commutative Property a. For Union - AUB b. For Intersection - AQB c. Idempotent — A U A and A n A d. Identity Law - A Of —A and A nt_J =A e. Set Difference is not commutative - 2. Associative Property For any three sets A, B and C a. AU(BUC) -(AnB)nc b. The set difference in general is not associative i. (A-B)-CI A-(B-C). c. If the sets A, B and C are mutually disjoint then the set difference is associative that is, (A—B)—C=A—(B—C). 3. Distributive Property a. For any three sets A, B and C i. An (BUC) [Intersection over union] ii. AU(BQC) [Union over intersection] 8
  9. 4. De Morgan's Laws for Set Difference For any three sets A, B and C i. A-(BUC) ii. A-(BnC) 5. De Morgan's Laws for Complementation Let 'U' be the universal set containing finite sets A and B. Then ii. (An B)' = A' B' Cardinality of Sets l. 2. 3. 5. n (A) +11 (B) n(AnB) n(AnB) 6. U B U C) = n(C)- n(AnB) For overlapping sets, a. n(AUB) = n(AnB) b. n(AnB) = - n(AUB) For disjoint sets, a. n(AUB) = b. n(AnB) = n(U) = n(A') + n C)- n(An n(An b z c Let three sets A, B and C represent the students. From the Venn diagram, Number of students in only set A= a, only set B = b, only set C = c . Total number of students in only one set=(a +b +c) Total number of students in only two sets=(x +y +z) Number of students exactly in three sets= r Total number of students in atleast two sets = x + y +z + r Total number of students in 3 sets = (a + b + c + x + y + z + r) 9

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