Introductory Discrete Mathematics

range of a function is a subset of its codomain

the image of x and x is a preimage of y

rule that assigns to every element in X a unique element in Y

The set X is the domain of the function and the set Y is its codomain. If

THEOREM 0.1.2 (De Morgan’s Laws)

(a) (A ∩ B)c = Ac ∪ Bc.

(b) (A ∪ B)c = Ac ∩ Bc.

region in the rectangle (which represents the universal set) that is outside the ellipses that represent the three sets is the absolute complement of the union of these three sets.

THEOREM 0.1.1 (Distributive Laws)

(a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

(b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

A class C(X) of subsets of a set X is called a partition of X if (1) C(X) is pairwise disjoint, and (2) the union of the sets in C(X) is the set X

Two sets are disjoint if and only if their intersection is empty.

both set intersection and set union possess the associative property: (1) A ∩ (B ∩ C) = (A ∩ B) ∩ C and (2) A ∪ (B ∪ C) = (A ∪ B) ∪ C.