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Operations with sets

 

Designation of sets and their elements. Equality of sets.

Subset ( inclusion ). Sum ( union ) of sets.

Product ( intersection ) of sets. Difference ( complement ) of sets.

Symmetric difference of sets. Properties of operations with sets.

 

Sets are designated by capital letters, and their elements by small letters. The record  a R  means, that an element    belongs to a set R, i.e.    is an element of the set R . Otherwise, if    doesn't belong to the set  R , we write  a R

 

Two sets and B are called equal ( = ), if they consist of the same elements, i.e. each element of the set  A is an element of the set B and vice versa, each element of the set    is an element of the set  A .

 

We say, that a set   is included in a set ( Fig.1 ) or the set  A is a subset of the set B  ( in this case we write   ), if each element of the set A is an element of the set B . This dependence between sets is called an inclusion. The inclusions    and     take place for each set  A .   

 

A sum ( union ) of sets  and ( it's written as ) is a set of elements, each of them belongs either to A, or to B. So, , if and only if either  , or   .  

 

A product ( intersection ) of sets  and ( it's written as    , Fig.2 ) is a set of elements, each of them belongs both to  and to . So,   , if and only if    and   .  

 

A difference of sets and ( it's written as  , Fig.3 ) is a set of elements, which belong to the set  A, but don't belong to the set  . This set is called also a complement of the set B  relatively the set A.

 

 

A symmetric difference of sets  A  and  B  ( it's written as  \ ), is called a set:

 

\ = ( A B ) ( A ) .

 

Properties of operations with sets:

 

 

 

E x a m p l e s.    1.  A set of children is a subset of the whole population.

 

2.  An intersection of the set of integers and the set of positive

                               numbers is the set of natural numbers.

 

3.  A union of the set of rational numbers and the set of irrational    

                               numbers is the set of real numbers.

       

                          4.  Zero is a complement of the set of natural numbers relatively

                               the set of non-negative integers.

 

 

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