Mathematics PRINCIPLE OF MATHEMATICAL INDUCTION

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PRINCIPLE OF MATHEMATICAL INDUCTION

Relation

1. Cartesian Product :

The Cartesian product of two sets `A, B` is a non-void set of all ordered pairs `(a, b)`,
where `a in A` and `b in B`. This is denoted by `A xx B`

`:.` `A xx B` = { `(a,b) | a in A` and `b in B` }
e.g. `A = { 1 ,2 } , B = {a,b}`
`A xx B = {(1,a) ,(1,b) ,(2,a),(2,b) }`

Note: (i) `A xx B ne B xx A` (Non-commutative)
(ii) `n (A xx B) =n (A) n (B) ` and `n( P (A xx B)) = 2^(n (A) n(B) )`
(iii) `A = phi` and `B= phi <=> A xx B = phi`
(iv) If `A` and `B` are two non-empty sets having n elements in common then `(Ax B)` and `(B x A)` have
`n^2` elements in common.
(v) `A xx ( B cup C ) = (A xx B) cup (A xx C)`
(vi) `A xx (B cap C) = (A xx B) cap ( A xx C)`
(vii) `A xx (B -C) = (A xx B) - (A xx C)`

Statistics

1. Definition :

STATISTICS : A set of concepts, rules and procedures that help us to:
--> Organize numerical information in the form of tables, graphs and charts;
--> Understand statistical techniques underlying decisions that affect our lives and well-being; and
--> Make informed decisions.

Data :

Facts, observations and information that come from investigations.
Generally three types of data are used

(i) Ungrouped data, Raw data or individual series:

(ii) Discrete frequency or ungrouped data :

Definition:

Data consist of `n` distinct values `x_1,x_2 ...... , x_n` occuring with frequency `f_1 ,f_2 , ...... , f_n` respectively.
This data in tabular form is called discrete freqency distribution.

(iii) Continuous frequency or grouped data :

Definition:
A continuous frequency Distribution is a series in which the data are classified into different class intervals
without gaps along with their respective frequencies.

Measures Of Central Value :

Measure of central value gives rough idea about where data points are centred. Mean, mode, median
are three measure of central tendency.

(A) Mean:

The mean is the most common measure of central tendency and the one that can be mathematically
manipulated. It is defined as the average of a distribution is equal to the `sumX /N`. Simply, the mean is
computed by summing all the scores in the distribution `(sumX)` and dividing that sum by the total number of
scores `(N)`.

Arithmetic mean of individual series (Ungrouped data) :

If the series in this case be `x_1, x_2, x_3 ...... , x_n` ; then the arithmetic mean `bar(x)` is given by

i.e., `bar(x) =` (sum of the series)/(Number Of terms) `= ( x_1 +x_2+x_3+...........+x_n)/N =1/N sum_(i=1)^(n) x_i`

Arithmetic mean for discrete frequency distribution :

If the terms of the given series be `x, x_2, ....... , x_n` and the corresponding frequencies be `f_1, f_2, ....... , f_n`,
then the arithmetic mean `bar(x)` is given by,

`bar(x) = (f_1x_1+f_2x_2+ ...............+ f_nx_n)/N = 1/N sum_(i=1)^(n) f_ix_i`. `( sum_(i=1) ^(n) f_i=N)`

Arithmetic mean for grouped or continuous frequency distribution :

Arithmetic mean `(bar(x)) =A +1/N sum_(i=1)^(n) f_i (x_i -A)`,

where `A =` assumed mean, `f =` frequency and `x - A =` deviation of each item from the assumed mean.

Combined Arithmetic mean :

If ` bar(x_i) (i =1 , 2, ...... , k)` are the means of `k`-component series of sizes `n_i , (i =1, 2, .. , k)` respectively, then
the mean `x` of the composite series obtained on combining the component series is given by the formula

`bar(x) = (n_1 bar(x_1) +n_2 bar(x_2) + ................+n_k bar(x_k))/(n_1+n_2+..........+n_k) = (sum_(i=1)^(n) n_(i) bar (x_i)) /(sum _(i=1) ^(n) n_i)` .

Weighted Arithmetic Mean :

Weighted arithmetic mean refers to the arithmetic mean calculated after assigning weights to different
values of variable. It is suitable where the relative importance of difference items of variable is not same.
Weighted Arithmetic Mean is give by

`bar(X_w) = ( sum_(i=1)^(n) W_i X_i) /(sum_(i=1) ^(n) W_i)`