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Permutation

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for the permutation we first need to understand factorial.  n!(read as n factorial)=n(n-1)!=n(n-1)(n-2)!= n(n-1)(n-2)(n-3)! in simple words n factorial is the multiplication of all previous numbers. for example: n=5 n!=5!=5*4*3*2*1=120 rules of factorial: 0!=1 1!=1 Permutation Permutation is an arrangement of things or characters or digits or anything which we have. In simple words, permutation is a number of arrangements that a number of things can have. Formula to find permutations: P(n,r) = npr = n!/(n-r)!    { Here, n!=n(n-1)!=n(n-1)(n-2)!=n(n-1)(n-2)(n-3)!} rules to calculate permutation: P(n,0)=nPo=1 P(n,1)=nP1=n P(n,n-1)=nP(n-1)=n For example: What are the different arrangements possible for the first 5 digits when we need only 4 digits at a time. 5P4=5!/4!=(5×4!)/(4!)=5 1. How many words can be made from the first 10 alphabets(capital) of lengh 6? solution: Here n=10 r=6 Total words from the alphabets are: 10P6=10!/6!=10×9×8×7×6!/7!= How many permutations are possible for 4 alp
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Basics of Counting

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Sum Rule If sets A and B are disjoint sets(the sets which have no common element in them), then  |A⋃B|=|A| + |B|  For example: a class has 43 girls and 54 boys total of the class=43+54=97 Product Rule If set A and set B the cartesian product is : A*B={(a,b)| a  ∈ A and b  ∈ B} if |A| =n and |B| = m then |A*B|=n*m for example: A={a,b} and B={c,d}   A*B={(a,c),(a,d),(b,c),(b,d)}  Examples on counting: 1. how many password satisfy the following requirements: it should be 6 to 8 characters long. starts with a letter. case sensitive. other characters: digits or letters. solution: first we will find the six character password as: =52*62*62*62*62*62 { here first 52 is for 26 small letters and 26 for capital letters, other 5 times 62 are for digits(10) and letters(52=26+26)} =52*62⁵ second for seven characters password as: =52*62* 62*62*62*62*62{ here one extra 62 is multiplied according to the 1st case it is for the seventh characters} = 52*62⁶ last for eight characters password as: =52*62*62
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what is Relation, reflexive relation, symmetric relation, asymmetric relation, transitive relation, equivalence relation, partial order relation

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 The relation is an association between two or more things. relation between sets is the association between then that defines how the sets are related.  Types of relations: Reflexive relation If a set A have elements like {a,b,c,d} the relation, must contain all (a,a), (b,b),(c,c),(d,d) in it then the relation is called as reflexive relation. in simple words the relation which contain (a,a) pair for all a n=belongs to the set, then the relation  on the set is called as reflexive relation. each element present in the set must be related to it self then the relation is called as reflexive relation. for example: A={1,2,3,6,8} the relation on set A is such that it defines (a,b) belongs to A when b is divisible by a. here R={(1,1),(1,2),(1,3),(1,6),(1,8),(2,2),(2,6),(2,8),(3,3),(3,6),(6,6),(8,8)}  here in the above relation all pairs that are like (a,a) are present in relation R that are: (1,1),(2,2),(3,3),(6,6),(8,8) all are present in the relation set so the above relation on set A is re
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Directed graph(diagraph)Hasse Diagram Discrete mathematics

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  directed graph A graph of the given relation. Let A and B are the two finite sets and R is the relation from A to B. For a graphical representation of a relation on a set is represented by a point. Each element in a set is represented by a point in diagraph. Those points are called as nodes. An edge is drawn from one point to another related point. This edges are called as arc. The direction in the diagraph is represented by an arrow. And all arrows in directed graph with atrows called as directed arcs. For example: Draw a  directed graph that represents the relation: R={(1,1),(2,2),(1,2),(2,3),(3,2),(3,1),(3,3)} The loops present in the diagraph represents the reflexive property that are (1,1),(2,2),(3,3) the edge from 1 to 2 represents (1,2) and similarly other edges are shown. Hasse diagram  Hasse diagram is a diagram which represents partial ordered set in the form of graph without showing its transitive relation. Steps to draw a Hasse diagram: Start with a directed graph  Remove
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what is Mathematical Induction, solved problems on Mathematical Induction

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 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) RHS=(k³+1+3(k)
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Preposition and logical connectivities in Discrete Mathematics

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 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 either a true value or false values. Connectives in prepositions Disconjunction/OR denoted by '∨'  conjunction/AND denoted by '∧' Not/Negation denoted by '∾' if then denoted by '→' if and only if '↔' Disconjuction/OR It combines two statements using or. Disconjunction is denoted by  '∨'. the
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