Definition : Sets
A set is an unordered collection of objects.
The objects in a set are called the elements, or members,of the set. A set is said to contain its elements.
Two sets are equal if and only if they have the same elements. That is, if A and Bare sets, then A and B are equal if and only if ∀x(x∈A↔x∈B). We write A=B if A and B are equal sets.
The set A is said to be a subset of B if and only if every element of A is also an element of B. We use the notation A⊆B to indicate that A is a subset of the set B.
Let S be a set. If there are exactly n distinct elements in S where n is a non negative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|.
A set is said to be infinite if it is not finite.
The Power Set
Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is
denoted by P(S).
denoted by P(S).
Cartesian Products
The ordered n-tuple (a1,a2,...,an) is the ordered collection that has a1 as its first element, a2 as its second element,...,and an as its nth element.
Let A and B be sets. The Cartesian product of A and B, denoted by A×B, is the set of all ordered pairs (a,b), where a∈A and b∈B. Hence, A×B={(a,b) | a ∈ A ∧ b ∈ B}.
The Cartesian product of the sets A1,A2,...,An, denoted by A1×A2× ··· ×An, is the set of ordered n-tuples (a1,a2,...,an), where ai belongs to Ai for i = 1,2,...,n. In other words,
A1×A2× ··· ×An = {(a1,a2,...,an) | ai ∈ Ai for i =1,2,...,n}.
