Mathematics

 Mathematical induction is a mathematical technique to prove statements. mathematical induction involves two steps: Basic step: to prove the given statement is true for the initial value. (f(0) is true) induction step: to assume statement true for k ( F(k) is true) and then prove it is true for F(k+1) then p(n) is true for natural numbers. Examples: Prove that 5ⁿ - 1 is divisible by 4 for all n>=1 solution: step-1: basic step : true for n=1 F(n)=5ⁿ - 1  F(1)=5¹-1 F(1)= 4 F(n) is true for n=1 step 2: Induction step first, we will assume the given statement is true for value k i.e. 2. Prove that n³ + 2n is divisible by 3 for all n>=1. solution: step-1: basic step : true for n=1 F(n)=n³ + 2n F(1)=(1)³+2(1) F(1)=1+2 F(1)=3 which is divisible by 3. hence the given statement is true for n=1. step 2: Induction step first, we will assume the given statement is true for value k i.e. F(k): k³-2k=3m is true to prove :  F(k+1): (k+1)³+2(k+1)=3m is true proof: RHS=(k+1)³+2(k+1) ...

  Venn diagram is a graphical representation of sets . Or a pictorial representation of sets represented by the closed figure is called a set diagram or Venn Diagram . Venn diagrams are used to illustrate various operations like union, intersection, and difference, the complement of a set. We can explain the relationship among sets through this in a more significant way. Venn diagram consists of: A universal set by using a rectangle. A set or subset of a universal set is represented by using a circle or ovals. Examples Venn diagram for set union: Venn diagram for set intersection: Venn diagram for complement of set A: Venn diagram for the complement of set B Example:1      In a survey of students, 50 students like Mathematics, 30 people like English, 10 people likes both             English and Mathematics. find the total number of students in the survey? solution:  Given data,      Student who like Mathematic...

The set in mathematics is the group of objects and the objects should be distinct. Repetition of objects is not allowed in sets.    The sets are used to define the concepts of relations and functions. the theory of sets is developed by German mathematician Georg Cantor (1845-1918). Definition of Set A set is any collection of distinct in simple way set is a collection of objects specified in such a way that we can determine whether a given object is present in the set or not.  for example : 1. A set of vegetables      2. A set of numbers 3. Set of flowers  4. A set of keys    Elements of a set  The different objects or members present in the set are called as elements of set. The elements of a set are written in any order . the elements present in the set should not be repeated. the elements are denoted by small letters.  Notation  A set is usually denoted by capital letters and the elements of the set are denoted by sma...