Sets

- Set: collection of distinguishable objects, where set haven’t same object twice.
Ex. {1,2,3}.
* Note.
 - If set have object more than once it called “ multiset ” .

- Two sets A, B are equal .. if they have the same elements : {1,2,3} = {2,3,1} = {3,2,1}.
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- Special sets :
Φ = empty set.

The empty set is a subset of every set, and every set is a subset of itself :
( ∅ ⊆ A )   , (  A ⊆ A ) .

Z = set of integers {.., -1, 0, 1, ... } .

N = set of natural numbers {0 ,1 ,2 ..} (Modern style) or {1, 2, 3 ..} ( Old style ).

R = set of real numbers.

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All sets under consideration are subsets of larger set (U) ..
Complement of set A as Ᾱ= U - A.

à    A∩Ᾱ=Φ      ,     A∪Ᾱ=U. 

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De morgan’s laws:

A- (B ∪ C) = (a - b) ∩ (a - c)     à Ᾱ∩B̄ = (A∪B)̅
A- (B∩C) = (a - b) ∪ (a - c)       à Ᾱ∪ B̄ = (A∩B)̅

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- A, B are disjoint if they have no elements in common A∩B=Φ.

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Cardinality: is a size of set |S| :

-        If cardinality of set is a natural number (N) , then set is finite else set is infinite.

-       Infinite set that can put into one to one correspondence with a natural number N is countable infinite else it is un countable.

Ex.

Z is countable, R is uncountable.

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Finite set of n-elements is sometimes called n-set... a 1-set is called singleton.

Subset of k-elements of set is sometimes called k-subset.

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Power of set S (2^S )

Set of all subsets, including empty set and set itself.
ex. 2^{{a, b}} ={Φ, {a}, {b},{a, b}} , it has a 2^|S| cardinality.

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Ordered pair of two elements a,b is denoted as (a, b).

It defined formally as (a, b)={a, {a, b}} and it isn’t equal (b, a).

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Cartesian product of 2 sets A and B denoted as A × B:

Set of all ordered pairs that its first element of the pair is element of A and the second is element of B.
A × B ={(a, b) : aϵ A and b ϵ B}.

Ex.
{a, b} ×{a, b, c} = {(a, a),(a, b),(a, c),(b, a),(b, b),(b, c)}
Note..

- |A×B| = |A| . |B|

- Cartesian product of n sets A_1,A₂ … A_n is a set of n-tuples :

A₁ × A₂ ×… A_n = {( a₁, a₂ …a_n ) : a_iϵA_i , i =1, 2, 3…n}.

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داليا مدكور
16/7
12:10