##### Basics of set theory | Set | Power Set | Cartesian Product

Set - A collection of unordered and distinct objects.
Examples:
N = {0, 1, 2, 3, .....} , the set of natural numbers
Z = {....., -2, -1, 0, 1, 2,....}, the set of integers
Z+ = {1, 2, 3,......}, the set of positive integers
Q = { p/q | p ε Z , and q≠0}, the set of rational numbers
Q+ = { x ε R | x=p/q, for some positive integers p and q }, the set of all positive rational numbers
R, the set of all real numbers

Power Set:
Given a set S, the power set of S is the set of all subsets of the set S. It is denoted by P(S).
Examples:
S = {0 ,1 ,2}
P(S) = P({0, 1, 2}) = {Φ, {0}, {1}, {2}, {0 ,1}, {0, 2}, {1, 2}, {1, 2, 3}}.
Note:
S = Φ
P(S) = {Φ}
P(P(S)) = {Φ, {Φ}}
Cartesian Product: The cartesian product of sets A and B, denoted by A x B, is the set of all ordered pairs (a, b), where aε A and b ε B. Hence,
A x B = {(a, b) | a ε A ∧ b ε B}
Example:
A = {1, 2}
B = {a, b, c}
A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

Set Identities: