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The history of set theory is rather different from the history of most other areas of mathematics. For most areas a long process can usually be traced in which ideas evolve until an ultimate flash of inspiration, often by a number of mathematicians almost simultaneously, produces a discovery of major importance.

Set theory however is rather different. It is the creation of one person, Georg Cantor. Before we take up the main story of Cantor's development of the theory, we first examine some early contributions.

Welcome to the set tutorial.
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Below you will see a list of this unit's page headings and the subjects discussed in each of these pages.
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Basics of Set

Subjects to be Learned


Definition (Equality of sets): Two sets are equal  if and only if  they have the same elements.
More formally,   for any sets A and B,  A = B   if and only if   x [ x A      x B ] .

Thus for example {1, 2, 3} = {3, 2, 1} , that is the order of elements does not matter, and {1, 2, 3} = {3, 2, 1, 1}, that is duplications do not make any difference for sets.

Definition (Subset): A set A is a subset of a set B if and only if   everything in A is also in B.
More formally,   for any sets A and B,  A is a subset of B, and denoted by A B,   if and only if   x [ x A      x B ] .
If A B, and A B, then A is said to be a proper subset of B and it is denoted by A B .

For example {1, 2} {3, 2, 1} .
Also {1, 2} {3, 2, 1} .

Definition(Cardinality): If a set S has n distinct elements for some natural number n, n is the cardinality (size) of S and S is a finite set. The cardinality of S is denoted by |S|.

For example the cardinality of the set {3, 1, 2} is 3.

Definition(Empty set): A set which has no elements is called an empty set.
More formally,  an empty set, denoted by , is a set that satisfies the following:
  x   x ,
where means "is not in" or "is not a member of".

Note that and {} are different sets. {} has one element namely in it. So {} is not empty. But has nothing in it.

Definition(Universal set): A set which has all the elements in the universe of discourse is called a universal set.
More formally,  a universal set, denoted by U , is a set that satisfies the following:
  x   x U .

Three subset relationships involving empty set and universal set are listed below as theorems without proof. Their proofs are found elsewhere .

Note that the set A in the next four theorems are arbitrary. So A can be an empty set or universal set.

Theorem 1: For an arbitrary set A   A U .

Theorem 2: For an arbitrary set A   A .

Theorem 3: For an arbitrary set A   A A .

Definition(Power set): The set of all subsets of a set A is called the power set of A and denoted by   2A  or   (A) .

For example for A = {1, 2},  (A) = { , {1}, {2}, {1, 2} } .

For B = {{1, 2}, {{1}, 2}, } ,  (B) = { , {{1, 2}}, {{{1}, 2}}, {}, { {1, 2}, {{1}, 2 }}, { {1, 2}, }, { {{1}, 2}, }, {{1, 2}, {{1}, 2}, } } .

Also   () = {}   and   ({}) = {, {}} .

Theorem 4: For an arbitrary set A,  the number of subsets of A is 2|A| .

Test Your Understanding of Basic Set Concepts