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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.
Set: A set is a well defined collection of objects.
Element: An object that belongs to a set is an element of the set. The notation for 'e is an element of the set S' is:
The opposite statement 'e is not an element of the set S':
Universal quantifier: ∀ means 'for all'.
Existential quantifier: ∃ means 'there exists'.
Explicit definition: lists the objects of the set. Example notation for set A containing elements H, He, Li:
Implicit definition: uses some conditions to define the elements of the set.
Using the previous example, define the same set as the 'list the chemical elements with atomic number smaller than 4'.
The conditions are listed after the symbol '|' which is read 'such that'.
Read it like this: the set of atomic species e such that for all e the atomic number is smaller than 4.
Venn diagram: pictogram that symbolizes a set.
Empty Set: the set that has no elements.
Subset: a set S is a subset of the set A if and only if all the elements of S are also elements of A.
The empty set is a subset of any set.
Operations with Sets
Set Equality: two sets are equal if and only if they are subsets of one another:
Set Union: all the elements that belong either to one set or to the other.
Set Intersection: all the elements that belong to both sets.
Disjoint Sets: two sets whose intersection is the empty set.
Set Difference: the subset of the elements of set A that are not elements of set B.
From all the definitions above, it is quite clear that each element of a set is distinct from all the other elements. In other words, there is no repetition of the same element.
This is an important difference between the definition of a set and the notion of 'data set' as used in statistics. In statistics, as we summarize the results of various experiments, there are experiments that may produce the same result. The identical results are recorded separately in the data set. Implicitly, the results do differ since they were recorded in different conditions (at different times or in different places) - however, at first sight a data set may look like this:
This does not constitute a set as defined by set theory. It is a set only if we introduce other data that disambiguates the meaning of the identical numbers.
This observation is important when we count the number of elements of a set.
Example: What is the union of the sets:
Notice how, in the union, we have only 7 elements: