Mathematical induction is a method of mathematical proof to used to establish that a given statement is true of all natural numbers. It consist of two step , I- Basis step and II- Induction Step. In First step we check the statement is true for n= 1 and when it is true them we proceed to step no-2 , induction step where we have to prove that given statement is true for n+1.
The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
The basis (base case): showing that the statement holds when n = 0 or n = 1.
The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.
The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.
Suppose the sum of first n natural number is given by : 1+2+3+….+n = n(n+1)/2
In this statement we have to prove first this statement is true for n=1 and then n= n+1 in induction step.
Sunday, June 28, 2009
Wednesday, June 24, 2009
What is cartesian product in Set Theory ?
In Set Theory, a cartesian product is the direct product of two sets call X and Y. It is denoted by X*Y , is the set of all possible ordered pairs whose first element is a member of X and Second element is from the Y.
A Cartesian product of two finite sets is represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.
The cartesian product is named after Rene Descartes.
For example - two coins are tossed whose possibilities are Head ( H) and Tail ( T) then the cartesian product of X and Y is
X*Y = ( H,T) , ( H,H) , ( T, H) , ( T,T)
Keep reading
A Cartesian product of two finite sets is represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.
The cartesian product is named after Rene Descartes.
For example - two coins are tossed whose possibilities are Head ( H) and Tail ( T) then the cartesian product of X and Y is
X*Y = ( H,T) , ( H,H) , ( T, H) , ( T,T)
Keep reading
Wednesday, June 17, 2009
Poisson distribution - A discrete Pobability distribution
Simeon Desnis Possion was the person who discovered this distribution. This is also pronounced as "Pwason". This probability distribution expresses the probability of a no. of events occuring in a fixed period of time if these events occurs with a average rate and independently of the time since the last event happened.
This distribution is mainly used for the number of events in a specified intervals such as distance , area or volume.
The fomula given my Possion is as :
f ( k ; y) = y power k x e power -y / K !
where -
e is the base of natural logarthim (e = 2.71828...)
k is the number of occurrences of an event - the probability of which is given by the function
k! is the factorial of k
y is a positive real number , equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute , and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with y = 10×4 = 40.
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms or the no. of accidents happened due to the fallen from roof in a metro city.
This distribution is mainly used for the number of events in a specified intervals such as distance , area or volume.
The fomula given my Possion is as :
f ( k ; y) = y power k x e power -y / K !
where -
e is the base of natural logarthim (e = 2.71828...)
k is the number of occurrences of an event - the probability of which is given by the function
k! is the factorial of k
y is a positive real number , equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute , and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with y = 10×4 = 40.
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms or the no. of accidents happened due to the fallen from roof in a metro city.
Tuesday, June 16, 2009
Binomial Theorem.. the expansion of a series
In Mathematics, the binomial theorem is an important formula giving the expansion of powers of sums and its simplest formula is as follows -
For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by:
(a+b) power n = a power n + na power n-1xb + n(n-1)/2 a power n-2 x b power 2 +.......+ b power n
By this formula , any series can be expanded.
This formula is given by the Blaise PAscal who described that in 17th century but The work on the formula had been started before Pascal gave it to the world.
Now try to expand the (x+y) power 10 by this formula.
For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by:
(a+b) power n = a power n + na power n-1xb + n(n-1)/2 a power n-2 x b power 2 +.......+ b power n
By this formula , any series can be expanded.
This formula is given by the Blaise PAscal who described that in 17th century but The work on the formula had been started before Pascal gave it to the world.
Now try to expand the (x+y) power 10 by this formula.
Wednesday, June 10, 2009
Creativity ...Separates the common man
Do you know ?
The Bell was invented by an Astronomer - Edmund Halley.
The Pneumatic tyre was invented by a veterinarian - John Dunlop
The safety razor was invented by a sales man - KC Gillete
The vacumm cleaner was invented by a bridge builder - Hubert Booth
There are other many example of the people who think out of box and gave the world new things. How? Because they were the creative people.
Who are the creative prople? Some characterstics are as follows-
•-Optimistic about future
•-Open to alternatives
•-Day dreamers
•-Highly curious and observant
•-Independent thinkers
•-Good at turning innovative ideas into practical solutions
•-Take action and make things happen
•-Adventurous with multiple interest
The Bell was invented by an Astronomer - Edmund Halley.
The Pneumatic tyre was invented by a veterinarian - John Dunlop
The safety razor was invented by a sales man - KC Gillete
The vacumm cleaner was invented by a bridge builder - Hubert Booth
There are other many example of the people who think out of box and gave the world new things. How? Because they were the creative people.
Who are the creative prople? Some characterstics are as follows-
•-Optimistic about future
•-Open to alternatives
•-Day dreamers
•-Highly curious and observant
•-Independent thinkers
•-Good at turning innovative ideas into practical solutions
•-Take action and make things happen
•-Adventurous with multiple interest
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