binomial theorem, recursion ,tower of honai, relations
DESCRIPTION
These slides contain proof of Binomial theorem , Tower of Honai , recursion a d properties of relation...TRANSCRIPT
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STARTS WITH THE NAME OF ALLAH
the most beneficent & Merciful
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PRESENTAION
AQEEL RAFIQUE 01
HAMZA MAQSOOD 12
REYAN IQBAL 28
UMAIR HAIDRI 17
SHOAIB ASHRAF 44
RAJA AMIR 43
PRESENTED BY :
GROUP MEMBERS :
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Binomial Theorem
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Binomial :In algebra (a+b) is called binomial. Binomial theorem provides an expression for the power of binomial of n.
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Binomial TheoremThe expansion of binomial theorem is.
an++bn
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Need of Binomial TheoremAs we know (a+b)0 =1 (a+b)1=(a+b)(a+b)2 =a2+2ab+b2
.
.(a+b)n=?
Binomial Theorem
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Binomial Theorem
= 1Let (1) is true for n=m
If true for n=m then also true for n=m+1
(1)Proof by Algebraic method
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Binomial TheoremR.H.S
=(a + b) =a .+b . =+
Replacing variables:Let j=k+1; k=j-1 when k=0, j=1
when k=m, j=m+1
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Binomial TheoremTaking second summation on the right hand side.
====
=am+1++bm+1
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Binomial Theorem by Pascal’s formula
Hence proved.This shows that if any number in the power of binomial is given we can easily find its expansion.
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Counting elements in one
dimensional array.
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Counting elements in one dimensional array.
Let A[1],A[2],A[3]……………….A[n] is a one dimensional array. Where n a positive integer.
To find the number of element in one dimensional array by using the theorem.
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Counting elements in one dimensional array.
We use theorem of find the no of elements in a list.i-e:- n-m+1
where n is the last term of the list and m is the first term of the list.
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Counting elements in one dimensional array.
Example:suppose the elements in 1 dimensional
array are;A[2]=2;A[3]=3;A[4]=5;...A[10]=7
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Counting elements in one dimensional array.
By Applying theorem we getApply theorem on index;
Where n=10, m=2;The number of elements in the given array are:
n-m+1=10-2+1=9
Elements = 9
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Recursion
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RecursionRecursively Defined Sequence
Method of defining a sequence: Informal ways Explicit formula Recursion
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Informal way: In informal ways a sequence is given
we extract or generate the pattern of the sequence and generate the next term. Disadvantages:
• Misunderstand of the sequence cause the error.
For example:if the sequence 3,5,7……. Is given
if some one guess it prime number place 9 if someone understand it prime number he put 11. so this cause the misunderstanding.
RecursionRecursively Defined Sequence
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Explicit formula: In explicit formula we make a formula
for the nth term of the sequence.For example: 2,4,6……….Explicit formula for above sequence is 2k, where k>0 Advantages:
• Each term is uniquely determine. Disadvantage:
• Difficult to make the explicit formula if such sequence is given which is difficult to analyze.
RecursionRecursively Defined Sequence
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Recursion:In recursion two equations are given.
• Recurrence relation:It is the formula to find the sequence.
• Initials conditions:it is the first few values of the sequence. It
is also called base or bottom of the recursion.
RecursionRecursively Defined Sequence
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For example:1) bk = bk-1+ bk-2 recurrence
relation
2)b0=1, b1=3 initial conditions
RecursionRecursively Defined Sequence
Every founded value is used to find the next term of the sequence
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For example:A sequence c0, c1, c2, . . . recursively as follows:
For all integers k ≥ 2,(1) ck = ck−1 + k.ck−2 + 1 recurrence
relation(2) c0 = 1 and c1 = 2 initial conditions.
Find c2,c3.
RecursionComputing Terms of a Recursively Defined Sequence
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Solution:Putting k=2 since c1 = 2 and c0 = 1
c2 = c1 + 2c0 + 1 = 2 + 2·1 + 1 =5similarly for c3 we put k=3 and solve.
RecursionComputing Terms of a Recursively Defined Sequence
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Let a1, a2, a3, . . . and b1, b2, b3, . . . satisfy the recurrence relation that the kth term equals 3 times the (k − 1) term for all integers k ≥ 2
(1)ak = 3ak−1 and bk = 3bk−1
And the initial conditions are:a1=3, b1=1
Find a2, a3 and b2 ,b3
RecursionSequences That Satisfy the Same Recurrence Relation
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RecursionSequences That Satisfy the Same Recurrence RelationSolution:
a2 = 3a1 = 3·3 = 9a3 = 3a2 = 3·9 = 27
So the ‘a’ sequence is 3,9,27…….
b2 = 3b1 = 3·1 = 3b3 = 3b2 = 3·3 = 9
So the ‘b’ sequence is 1,3,9,………
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Tower of Hanoi
The Tower of Hanoi (also called the Tower of Brahma or Lucas' Tower, and sometimes pluralized) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.
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nn-1
1
A B C
Tower of Hanoi
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The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules:
• Only one disk can be moved at a time.• Each move consists of taking the upper disk from
one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack.
• No disk may be placed on top of a smaller disk.
ObjectiveTower of Hanoi
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Recursive pattern:From the moves necessary to transfer one, two, and three disks, we can find a recursive pattern - a pattern that uses information from one step to find the next step - for moving n disks from post A to post C:
First, transfer n-1 disks from post A to post B. The number of moves will be the same as those needed to transfer n-1 disks from post A to post C. Call this number M moves.
Next, transfer disk 1 to post C [1 move].
Finally, transfer the remaining n-1 disks from post B to post C. [Again, the number of moves will be the same as those needed to transfer n-1 disks from post A to post C, or M moves.]
No of moves : (n-1)+1+(n-1)
2(n-1)+1
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What?if we want to know how many moves it will take to transfer 100 disks from post A to post B.Ans: Through recursion we will first have to find the moves it takes to transfer 99 disks, 98 disks, and so on.So now we find the explicitly:Number of Disks Number of Moves 1 1 2 3 3 7 4 15 5 31 The pattern generated from this sequence is: 2n-1
Tower of Hanoi
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Tower of Hanoi
1,3,7,15…………….1+0=1 21-1=11+2=3 22-1=33+4=7 23-1=77+8=15 24-1=15. .. .. .So the minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n is the number of disks.
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Relations
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RelationsA relation R is a subset of the Cartesian product of the given set(s).
Given in order pair form (x , y).
x related to y, if and only if (x , y) is in R.
Denoted by x R y.
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RelationsProperties of Relations:
Reflexive Symmetric Transitive
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RelationsProperties of Relations:
Let R be a relation on set A Reflexive:
R is reflexive if and only if x R x for all x is in A.
Symmetric:R is symmetric if x R y then y R x ; x , y is in A
Transitive:R is transitive if x R y and y R x then x R z; x , y , z is in
A
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RelationsProperties of Relations:
Let A = {0, 1, 2, 3} and define relations R on A as follows:R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)},Is R reflexive, symmetric, transitive ?Solution:Graph of relation will be
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RelationsProperties of Relations:
Reflexive:R is reflexive because each element contain loop, mean each element is related to itselfSymmetric:R is symmetric because here an arrow move from one point to second and also from second to first, mean first related to second and also second related to first.Transitive:R is not transitive because there is no arrow moves from 3 to 1. so 1 R 0 and 0 R 3 but 1 R 3.
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Any Question?
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Thanks!