Tiêu đề Mathematics Handbook.pdf ✅

1 Sets and Relations Set Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by capital letters A B C , , ,K and elements are usually denoted by small letters a b c , , ,... . If a is an element of a set A, then we write a A ∈ and say a belongs to A or a is in A or a is a member of A. If a does not belongs to A, we write a A ∉ . Standard Notations N : A set of all natural numbers. W : A set of all whole numbers. Z : A set of all integers. Z Z + − / : A set of all positive/negative integers. Q : A set of all rational numbers. Q Q + − / : A set of all positive/negative rational numbers. R : A set of all real numbers. R R + − / : A set of all positive/negative real numbers. C : A set of all complex numbers. Methods for Describing a Set (i) Roster Form / Listing Method / Tabular Form In this method, a set is described by listing the elements, separated by commas and enclosed within braces. e.g. If A is the set of vowels in English alphabet, then A a e i o u = { , , , , } (ii) Set Builder Form / Rule Method In this method, we write down a property or rule which gives us all the elements of the set. e.g. A x x = { : is a vowel in English alphabet} Types of Sets (i) Empty/Null/Void Set A set containing no element, it is denoted by φ or { }.
(ii) Singleton Set A set containing a single element. (iii) Finite Set A set containing finite number of elements or no element. Note : Cardinal Number (or Order) of a Finite Set The number of elements in a given finite set is called its cardinal number. If A is a finite set, then its cardinal number is denoted by n A( ). (iv) Infinite Set A set containing infinite number of elements. (v) Equivalent Sets Two sets are said to be equivalent, if they have same number of elements. If n A n B ( ) ( ) = , then A and B are equivalent sets. (vi) Equal Sets Two sets A and B are said to be equal, if every element of A is a member of B and every element of B is a member of A and we write it as A B= . Subset and Superset Let A and B be two sets. If every element of A is an element of B, then A is called subset of B and B is called superset of A and written as A B ⊆ or B A ⊇ . Power Set The set formed by all the subsets of a given set A, is called power set of A, denoted by P A( ). Universal Set (U) A set consisting of all possible elements which occurs under consideration is called a universal set. Proper Subset If A is a subset of B and A B ≠ , then A is called proper subset of B and we write it as A ⊂ B. Comparable Sets Two sets A B and are comparable, if A B ⊆ or B A ⊆ . Non-comparable Sets For two sets A B and , if neither A B ⊆ nor B A ⊆ , then A and B are called non-comparable sets. Disjoint Sets Two sets A and B are called disjoint, if A B ∩ = φ. i.e. they do not have any common element. 2 Handbook of Mathematics
Intervals as Subsets of R (i) The set of real numbers x, such that a x b ≤ ≤ is called a closed interval and denoted by [ , ] a b i.e. [a, b] = { : , } x x R a x b ∈ ≤ ≤ . (ii) The set of real number x, such that a x b < < is called an open interval and is denoted by ( , ) a b i.e. ( , ) a b = { : , } x x R a x b ∈ < < (iii) The sets [ , ) { : , } a b x x R a x b = ∈ ≤ < and ( , ] { : , } a b x x R a x b = ∈ < ≤ are called semi-open or semi-closed intervals. Venn Diagram In a Venn diagram, the universal set is represented by a rectangular region and its subset is represented by circle or a closed geometrical figure inside the rectangular region. Operations on Sets 1. Union of Sets The union of two sets A and B, denoted by A B ∪ , is the set of all those elements which are either in A or in B or both in A and B. Laws of Union of Sets For any three sets A, B and C, we have (i) A A ∪ =φ (Identity law) (ii) U A U ∪ = (Universal law) (iii) A A A ∪ = (Idempotent law) (iv) A B B A ∪ = ∪ (Commutative law) (v) ( ) A B C A B C ∪ ∪ = ∪ ∪ ( ) (Associative law) Sets and Relations 3 U A U A B
2. Intersection of Sets The intersection of two sets A and B, denoted by A B ∩ , is the set of all those elements which are common to both A B and . If A A A 1 2 n , ,... , is a finite family of sets, then their intersection is denoted by ∩ ∩ ∩ ∩ i = n A A A A i n 1 1 2 or ... . Laws of Intersection For any three sets, A, B and C, we have (i) A ∩ =φ φ (Identity law) (ii) U A A ∩ = (Universal law) (iii) A A A ∩ = (Idempotent law) (iv) A B B A ∩ = ∩ (Commutative law) (v) ( ) A B C A B C ∩ ∩ = ∩ ∩ ( ) (Associative law) (vi) A B C A B A C ∩ ∪ = ∩ ∪ ∩ ( ) ( ) ( ) (intersection distributes over union) (vii) A B C A B A C ∪ ∩ = ∪ ∩ ∪ ( ) ( ) ( ) (union distributes over intersection) 3. Difference of Sets For two sets A and B, the difference A B − is the set of all those elements of A which do not belong to B. Symmetric Difference For two sets A B and , symmetric difference is the set ( ) ( ) A B B A − ∪ − denoted by A B ∆ . 4 Handbook of Mathematics A U A B U A B U A B A B – B A –