Title Mathematics Handbook.pdf ✅

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 –