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) ...

 Preposition A preposition is a declarative statement that is either true or false, but not both. Prepositions contain connectives and variables. The variables in the preposition are denoted by capital letters(A, B,....Z). for example: India is a state                                returns 'false' Maharashtra is a country.               returns 'false' 12-3=9                                            returns 'true' 10-4=7                                          returns  'false' not prepositions    A is greater than 12       x+3=6 go and playout these statements are not prepositions because the statements do not have...

A multiset is a set of elements in which every element can occur more than once. This is the different property of multiset from the sets. a set in which the occurrence of an element can be more than one or can be 0. The multiplicity of the element is the occurrence of the element in the multiset. the  multiplicity of an element can be 1 or more than one or can be 0. the size of the multiset is the sum of all the multiplicities of all the elements.  for example: A= {1,1,1,1,2,2,3,4,4,5} The multiplicity of element '1' in multiset A is 4. The multiplicity of element '2' in multiset A is 2. The multiplicity of element '3' in multiset A is 1. The multiplicity of element '4' in multiset A is 2. The multiplicity of element '5' in multiset A is 1. the size of the multiset A is 10 In sets, the cardinality of the set is the multiplicity of the set. and rest all things the representation of a set and multiset is similar. in multisets A={1,1,1,1,2,2,3,4,4...

 In this article, you will learn all about the Principle of inclusion and exclusion. the principle of inclusion and exclusion explains how to find the total number of elements that belong to the particular set. principle: for two arbitrary set A and B the principle states as : |A ᴜ B| = |A| + |B| – |A ∩ B| where |A| is the cardinality of the set A. cardinality of the set is the number of elements present in the set. for three sets the principle of inclusion and exclusion is symbolically represented as: |A ∪ B ∪ C| = |A| + |B| + |C| – | A ∩ B|  –  |A ∩ C|  –  |B ∩ C| + |A ∩ B ∩ C| Examples: example 1: Suppose a survey of 100 people asks if they have a cat or dog as a pet.  The results are as follows:  55 answered yes for the cat, 58 answered yes for dog and 20 people checked yes for  both cat and dog. How many people have a cat or a dog? solution: the people who have a cat is denoted by set C the people who have a dog is denoted by  set D |C|...