##### 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:
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