key concept
set: a set is an unordered collection of objects. (It is not part of a formal theory of sets)
This intuitive notation will lead to paradoxes, or logical inconsistencies.
naive set theory: not develop an axiomatic version of set theory.
set builder: the defination of belonging symbol
two sets are equal, A=B: if and only if they have the same elements. every element of A is also an element of B, and every element of B is also an element of A.
A is a subset of B: if and only if ervery element of A is also an element of B
A is a proper subset of B: if and only if A is a subset of B and A!=B
cardinality of set: if set A has finite number of elements, then the number is called cardinality of set A
the power set of A: the set of all subsets of A
cartesian product of A and B, A*B: is the set of all ordered pairs (a,b), where a belonging to A, and B belonging to B.
truth set of P: {x belongs to A | P(x)}, is a subset of A
union of A and B: {x| x belongs to A or x belongs to B}
intersection of A and B: {x| x belongs to A and x belongs to B}
difference of A and B, A-B: {x| x belongs to A and x doesn't belong to B}
complement of A: U-A, where U is the universal set
set identities: set equations no matter what set is, including distributive laws, De Morgan's law, and so on. They can be prooved by mathematic logic and definations of set
computer representation of sets: assume the universal set U is finite, set can be represented by bit string