1. Set

    는 Definition

set

an unordered collection of object.

A = {1, 3, 5, 7}

B = { n | n is a prime number} which we read as 'the set of all n such that n is a prime number'.

Two sets X and Y equal if ((x∈X -> x∈Y) ^ (x∈Y -> x∈X)).

cadinality, denoted by |X|, is the number of elements in a set X.(size)

subset

We write X⊆Y to mean that any member of X is also in Y, and say that X is a subset.

X is proper subset if X ⊆ Y and X≠Y. 

Power set

The set of all subsets of X is called its power set P(X). Thus

P(X) =  { Y | Y⊆X}

partition

A partition S on a set X is a familly {A1, A2, ... , An} of subsets of X, such that

S = {A| (A⊆X) ^ (Aj∩Ak = ø. for every j, k) ^ ( A1∪A2∪ ... ∪ An = X)

operation

①union

X∪Y = { x| x∈X or x∈Y}

② intersection

X∩Y = { x| x∈X and x∈Y}

Two sets X and Y are disjoint if X∩Y = ø.

A collection of sets S is said to be pairwise disjoint if whenever X and Y are distinct sets in S, X and Y are disjoint. 

③ difference

X - Y = { x| x∈X and x∉Y}

④ complement

X^￣ = U - X

⑤ Other Notation

Let S = {A1, A2, A3, ... , An}

∪S = { x|x∈X for some X∈S} = A1∪A2∪ ... ∪An

∩S = { x|x∈X for all X∈S} = A1∩A2∩ ... ∩An 

pairwise disjoint의 예) S = { { 1, 4, 5}, {2, 6}, {3}, {7, 8} }

Properties of set operations

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