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 à belongs to a set R , i.e. à is an element of the set R . Otherwise, if à doesn't belong to the set R , we write a R .
Two sets À and B are called equal ( À = Â ), 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 Â is an element of the set A .
We say, that a set À is included in a set Â ( Fig.1 ) or the set A is a subset of the set B ( in this case we write À Â ), if each element of the set A is an element of the set B . This dependence between sets is called an inclusion . The inclusions À and À À take place for each set A .
A sum ( union ) of sets À and Â ( it's written as À Â ) is a set of elements, each of them belongs either to A , or to B . So, å À Â , if and only if either å À , or å Â .
A product ( intersection ) of sets À and Â ( it's written as À Â , Fig.2 ) is a set of elements, each of them belongs both to À and to Â . So, å À Â , if and only if å À and å Â .
A difference of sets À and Â ( it's written as À – Â , Fig.3 ) is a set of elements, which belong to the set A , but don't belong to the set Â . 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 À \ Â ), is called a set:
À \ Â = ( A – B ) ( Â – 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.