Set theory

Important theorem

Sets

A = B if and only if A⊃B and A⊂B

There is one empty set ∅

∅⊂ All other sets

Power Sets
Gien a set A, its power set P(A) is the set of all subsets of A

This includes all combinations of the set members, plus the empty set, and the initial set

Ordered pair
Binary operations

Basic universal set operations
A∩𝒰 = A

A∪𝒰 = 𝒰

There are many correspondences between set theory and logic that can be seen as equivalents

For example compliment set and negation

Union as or

Intersection as and

Set Identities

 * 1) Commutative properties
 * 2) A∩B = B∩A
 * 3) A∪B = B∪A
 * 4) Associative properties
 * 5) A∩(B∩C) = B∩A
 * 6) A∪B = B∪A

Infinity
Natural numbers are denumerable or countably infinite because we can express them as a sequence

Cardinality

 * A| absolute value signs are used to indicate the number of members of a set