ICSE Class 8 Mathematics Sets PDF

ICSE - Class 8 - Mathematics Chapter 1: Sets (Lecture Notes) What is a Set? A set is a well-defined collection of distinct objects. – Example: A = {1, 2, 3, 4, 5} What is an element of a Set? The objects in a set are called its elements. – So in case of the abov...

ICSE - Class 8 - Mathematics Chapter 1: Sets (Lecture Notes) What is a Set? A set is a well-defined collection of distinct objects. – Example: A = {1, 2, 3, 4, 5} What is an element of a Set? The objects in a set are called its elements. – So in case of the above Set A, the elements would be 1, 2, 3, 4, and 5. We can say, 1  A, 2 A etc. Usually we denote Sets by CAPITAL LETTERs like A, B, C, etc. while their elements are denoted in small letters like x, y, z etc. If x is an element of A, then we say x belongs to A and we represent it as x  A If x is not an element of A, then we say that x does not belong to A and we represent it as xA How to describe a Set? Roaster Method or Tabular Form – In this form, we just list the elements – Example A = {1, 2, 3, 4} or B = {a, b, c, d, e} Set- Builder Form or Rule Method or Description Method – In this method, we list the properties satisfied by all elements of the set – Example A = {x : x  N, x < 5} 1 For more information please go to: https://icsemath.com/  Some examples of Roster Form vs Set-builder Form Roster Form Set-builder Form 1 {1, 2, 3, 4, 5} {x | x  N, x 100} 2 For more information please go to: https://icsemath.com/  Cardinal number of Finite Set  The number of distinct elements contained in a finite set A is called the cardinal number of A and is denoted by n(A) Example A = {1, 2, 3, 4} then n(A) = 4 A = {x | x is a letter in the word ‘APPLE’}. Therefore A = {A, P, L, E} and n(A) = 4 A = {x | x is the factor of 36}, Therefore A = { 1, 2, 3, 4, 6, 9, 12, 18, 36} and n(A) = 9 Empty Set A set containing no elements at all is called an empty set or a null set or a void set. It is denoted by ϕ (phai) In roster form you write ϕ = { } Also n (ϕ) = 0 Examples: {x | x  N, 3 < x 5} = ϕ Non Empty Set A set which has at least one element is called a non-empty set Example: A = {1, 2, 3} or B = {1} Singleton Set A set containing exactly one element is called a singleton set Example: A = {a} or B = {1} 3 For more information please go to: https://icsemath.com/  Equal Sets Two set A and B are said to be equal sets and written as A = B if every element of A is in B and every element of B is in A Example A = {1, 2, 3, 4} and B = {4, 2, 3, 1} It is not about the number of elements. It is the elements themselves. If the sets are not equal, then we write as A ≠ B Equivalent Sets Two finite sets A and B are said to be equivalent, written as A ↔ B, if n(A) = n(B), that is they have the same number of elements. Example: A = {a, e, i, o, u} and B = {1, 2, 3, 4, 5}, Therefore n(A) = 5 and n(B) = 5 therefore A ↔ B Note: Two equal sets are always equivalent but two equivalent sets need not be equal. Subsets If A and B are two sets given in such a way that every element of A is in B, then we say A is a subset of B and we write it as A ⊆ B Therefore is A ⊆ B and x  A then x  B If A is a subset of B, we say B is a superset of A and is written as B  A Every set is a subset of itself. i.e. A ⊆ A, B ⊆ B etc. Empty set is a subset of every set i.e. ϕ ⊆ A, ϕ ⊆ B If A ⊆ B and B ⊆ A, then A = B 4 For more information please go to: https://icsemath.com/  Similarly, if A = B, then A ⊆ B and B ⊆ A If set A contains n elements, then there are 2n subsets of A Power Set The set of all possible subsets of a set A is called the power set of A, denoted by P(A). If A contains n elements, then P(A) = 2n sets. i.e. if A = {1, 2}, then P(A) = 22 = 4 Empty set is a subset of every set So in this case the subsets are {1}, {2}, {2, 3} & ϕ Proper Subset Let A be any set and let B be any non-empty subset. Then A is called a proper subset of B, and is written as A B, if and only if every element of A is in B, and there exists at least one element in B which is not there in A. – i.e. if A ⊆ B and A  B, then A B – Please note that ϕ has no proper subset – A set containing n elements has (2n – 1) proper subsets. i.e. if A = {1, 2, 3, 4}, then the number of proper subsets is (24 – 1) = 15 Universal Set If there are some sets in consideration, then there happens to be a set which is a superset of each one of the given sets. Such a set is known as universal set, to be denoted by U or  i.e. if A = {1, 2}, B = {3, 4}, and C = {1, 5}, then U or  = {1, 2, 3, 4, 5} 5 For more information please go to: https://icsemath.com/  Operations on Sets Union of Sets The union of sets A and B, denoted by A  B, is the set of all those elements, each one of which is either in A or in B or in both A and B If there is a set A = {2, 3} and B = {a, b}, then A  B = {2, 3, a, b} So if A B = {x | x  A or x  B} , then x  A  B which means x  A or x B And if x  A  B which means x  A or x  B Interaction of Sets The intersection of sets A and B is denoted by A  B, and is a set of all elements that are common in sets A and B. i.e. if A = {1, 2, 3} and B = {2, 4, 5}, then A  B = {2} as 2 is the only common element. Thus A  B = {x: x  A and x B} then x  A  B i.e. x A and x  B And if x  A  B i.e. x A and x  B Disjointed Sets Two sets A and B are called disjointed, if they have no element in common. Therefore: A  B = ϕ i.e. if A = {2, 3} and B = {4, 5}, then A  B = ϕ Intersecting sets Two sets are said to be intersecting or overlapping or joint sets, if they have at least one element in common. Therefore two sets A and B are overlapping if and only if A  B  ϕ Intersection of sets is Commutative 6 For more information please go to: https://icsemath.com/  i.e. A  B = B  A for any sets A and B Intersection of sets is Associative i.e. for any sets, A, B, C, (A  B)  C = A  (B  C) If A ⊆ B, then A  B = A Since A ⊆ , so A   = A For any sets A and B, we have A  B ⊆ A and A  B ⊆ B A  ϕ = ϕ for every set A Difference of Sets For any two sets A and B, the difference A – B is a set of all those elements of A which are not in B. i.e. if A = {1, 2, 3, 4, 5} and B = {4, 5, 6}, Then A – B = {1, 2, 3} and B – A = {6} Therefore A – B = {x | x  A and x  B}, then x  A – B then x  A but x B If A  B then A – B =  Complement of a Set Let x be the universal set and let A  x. Then the complement of A, denoted by A’ is the set of all those elements of x which are not in A. i.e. let  = {1, 2, 3, 4, 5,6 ,7 ,8} and A = {2, 3,4 }, then A’ = {1, 5, 6, 7, 8} Thus A’ = {x | x   and x A} clearly x  A’ and x  A Please note ’ =  and ’=  7 For more information please go to: https://icsemath.com/  A  A’ =  and A  A’ = ϕ Disruptive laws for Union and Intersection of Sets For any three sets A, B, C, we have the following A  (B  C) = (A B)  (A  C) Say A = {1, 2}, B = {2, 3} and C = {3, 4} Therefore A  (B  C) = {1, 2, 3} and And (A  B)  (A  C) = {1, 2, 3} and hence equal A  (B  C) = (A  B)  (A  C) Say A = {1, 2}, B = {2, 3} and C = {3, 4} Then A  (B  C) = {2} and (A  B)  (A  C) = {2} and hence equal Disruptive laws for Union and Intersection of Sets De-Morgan’s Laws – Let A and B be two subsets of a universal set , then (A  B)’ = A’  B’ (A  B)’ = A’  B’ Let  = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3} and B = {3, 4, 5} Then A  B = {1, 2, 3, 4, 5}, therefore (A  B)’ = {6} A’ = {4, 5, 6} and B’ = {1, 2, 6} Therefore A’  B’ = {6}. Hence proven 8 For more information please go to: https://icsemath.com/