Set theory Chapter 1( class 11)

Symmetric difference of two sets 1) Meaning:-   A Δ B = (AUB) - (A∩ B)  because  in the venn diagram we can see that AUB is the entire A and B circle and from that if we remove A∩B( ∝,β part) , we get only A and only B which is AΔ B 2) Number of elements in AᅀB and a question to support that: Also n(AΔB) = n(A) + n(B) - 2n(A⋂B) This formula is also important. 3) Question:

  Difference of two sets 1) Meaning of difference between two sets Basically A-B means remove all elements from A that is also present in B  and B-A means remove all elements from B that is also present in A 2) Different Formulas to write A-B and B-A 3) Diagrammatic proof of why A-B = A∩B ᶜ The Red line region is B compliment and both Red line  +  White line  shaded region is  A∩B ᶜ. 4) Number of elements in A-B and B-A & a short Question:

Intersection of sets: 1) Meaning of intersection of sets   ** Intersection means the common elements, so the common element in all A, B and C is J. 2) Intersection of n sets symbol representation. 3) Some Basic properties of intersection of sets: 4) Some NCERT sums:

  Union of sets  1) Union meaning 2) Representation of union of n sets in symbol form: ** Here big U represents union of Ai sets, and Ai represents variables , where the value of i  starts from , i=1  and ends at n. Therefore the union of Ai will contain elements [A1 , An] , this is a closed interval meaning A1 and An are included 3) Venn-diagram representation of Union of sets: ** the entire shaded area is A union B 4) Union of three sets (** in the picture U set is universal set) 5) Some Properties of Union of sets: 6) Some NCERT sums:

  Complement of a set/ also known as negation or not-gate in Physics 1) Complement of a set definition 2) Euler-venn diagram representation of sets 3) Some questions on complement of sets 4) Some Important rules to remember on complement of sets

  Operations on sets 1) Complement of a set. 2) Addition / Union of two sets. 3) Intersection of two sets. 4) Difference of two sets. 5) Symmetric difference of two sets.

  Power set 1) what does power set mean => A set containing all the possible subsets of a particular set is called as its power set. P(A) is read as power set of A and n(P(A)) is read is number of elements of power set of A. 2) Question on Power set

  Symbol of subsets 1) ⊂ is the Symbol of subset For example:  A= {1,2,3}                         B={1,2}                         C={1} C ⊂  A B⊂ A 2) ⊆ is the symbol of subset or equal to. 3) Null set has only 1 subset and that is null set itself. { } ⊂ { }.

  Subsets 1) Subsets  => A set 'A' is called as subset of 'B' if and only if all elements of 'A' are in 'B'.     If 'A' is a subset of 'B', then 'B' is called as superset of 'A'.    Symbol of superset => B ⊇ A *** there is a mistake in the picture, null set is a proper subset of every set. ** This is read as, for every a that belongs to X if a belongs to Y too, then X is a subset of Y 2) Proper Subsets  => Proper Subset =( Total number of subsets - 1)  (or 2^n -1) for example in the above example the total subset is 8 , therefore proper subset is = 8-1 i,e 7. 3) Finding total number of subsets with the formula 2 ⁿ => If A is a set ,then the total number of subsets of A = 2 ⁿ⁽ᴬ⁾ 4) Some notes on subsets

  Intervals 1) Open interval and Closed interval Examples : 1. Open bracket (1,2) or ]1,2[ = x∊R : 1<x<2  ( 1 and 2 are not included) 2. Closed square brackets [1,2] = x ∊ R : 1<=x<=2 ( 1 and 2 are included) 2) Semi-open or Semi-closed interval

  Types of sets 1) Universal set => The set that has all the elements relevant to our question. 2) Real Number line => The real number line is a representation of all the real numbers on a horizontal line such that each point on the line corresponds to a real number and every real number corresponds to a point in the line. (-∞, ∞) : REAL Line.

  Types of sets 1) Equal set => If all the elements of two sets are equal, then they are called equal set. Example:- i) A= { 1,2}    B= { x : x^2 - 3x+2 =0 , x∊ R} finding the roots of x^2 - 3x+2 =0 by splitting the middle term method x^2- 3x+2 =0 x^2 - [2x+x] + 2 =0 x^2 -2x -x + 2=0 x(x-2) -1(x-2) =0  (x-2) (x-1) x= 2 , x=1 therefore we get to know that set A= set B ie, A={1,2} and B={1,2} 2) Equivalent set   => If cardinal no. of sets are equal , then they are said to be equivalent sets. Example:- C={1,2} & D={x,y} , n(C) = 2 and n(D) = 2 , therefore n(C)=n(D). Note:- All equal sets are equivalent sets.

  Types of set 1) Singleton set => if no. of elements in a set is '1' , it is called a singleton set. Examples: i) { w} ii) B= { x: x ² -1 = 0 , x >0} or in roaster form B= {1} iii) The cardinal number is 1 .

  Types of set 1) null / void / empty set => If number of elements in a set are zero , it is called a null set. It is represented by:- i) { } or ii) ɸ (phi) Example:  a) A={ x: x∊R , x ²+ 1 =0}, there is no such value of x such that x^2 + 1 = 0 b) cardinal number of null set is 0.

Types of sets( finite and infinite) 1) Finite and infinite sets.   2) Cardinal number of a set or order of a set. = Number of distinct elements in a set is known as a cardinal number of a set. it is represented in 3 ways : i) n(A) - read as no of elements of set A ii) O(A) - order of set A iii) |A| example:- A = {1,2,3,x,y} n(A) = 5 3) Standard Infinite sets i) N - set of all natural numbers ii) W - set of all whole numbers iii) R - set of all real numbers iv) R ⁺ - set of all +ve real numbers v) R ⁻ - set of all -ve real numbers vi) Q - set of all rational numbers vii) I , z - set of all integers viii) I+ve or z+ve - set of all +ve integers ix) I-ve or z-ve - set of all -ve integers x) R - Q => set of all irrational numbers

  Representation of sets: 1) Roaster or Tabular form => A={1,2,3,4,5} 2) Set Builder or property form => A= { x: 5>=x>=1 , x ∊ N} or this can be written as A={ x:x ∊ N, x<=5} This is read as = set A contains all  those x, such that x is greater than or equal to 1 and less than equal to 5 and x is a Natural number. In hindi it will be read as = set A meh woh sareh x hongeh jo 1 and 5 ke beech ke ho including 1 and 5 aur natural number bhi ho. 3) Visual Representation 4) Write A={2,4,6,8} in set builder form => A= {2n : n∊ N , n<=4}  2x1 = 2 2x2 = 4 2x3= 6 2x4= 8 5) Write the set A={1,4,9,16,25,....} in set builder form => A={ x ²: x∊N} 1 ² = 1 2 ²= 4 3 ²=9 4 ²=16 5 ²= 25 and so on 6) Write the set { x: x is a positive integer and x ³ <100} in roaster form => {1,2,3,4} because 1 ³ = 1< 100 2 ³ = 8 <100 3 ³ = 27<100 4 ³ = 64<100 5 ³ = 125 not less than 100 %3CmxGraphModel%3E%3Croot%3E%3CmxCell%20id%3D%220%22%2F%3E%3CmxCell%20id%3D%2...