Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 260

236 Logic as a Tool These properties of the powerset imply that, if U is a universal set in which all sets of our interest are included, then ∪, ∩, and are operations on P(U ) (We write A for U − A.) Thus, we obtain an algebraic structure P(U ); ∪, ∩, , ∅, U called the powerset Boolean algebra of U Proposition 191 (Properties of the Cartesian product) The following hold for any sets A, B, C (a) A × (B ∪ C ) = (A × B ) ∪ (A × C ), (B ∪ C ) × A = (B × A) ∪ (C × A) (b) A × (B ∩ C ) = (A × B ) ∩ (A × C ), (B ∩ C ) × A = (B × A) ∩ (C × A) (c) A × (B − C ) = (A × B ) − (A × C ) 5.2.3 Functions First, we recall the basic terminology: a function (or mapping) from a set A to a set B is a rule denoted f : A → B which assigns to each element a ∈ A a unique element f (a) ∈ B The element f (a) is called the value of a under f , or the image of a under f If f (a) = b then a is called a pre-image of b under f The set A is called the domain of f , denoted A = dom(f ), andB is called the co-domain, or the target set, of f , denoted B = cod(f ) The notion of image can be generalized from elements to subsets of the domain as follows For any subset X ⊆ A of the domain of f , the image of X under f is the set f [X ] = {f (a) | a ∈ X } The image of the whole domain A under f , that is, the set f [A] = {f (a) | a ∈ A} of all values of f , is called the range or image of f , also denoted rng(f ) Two functions f and g are equal iff dom(f ) = dom(g), cod(f ) = cod(g ), and f (a) = g (a) for every a ∈ dom(f ) The graph of a function f is the set of ordered pairs {(a, f (a)) | a ∈ dom(f )} A function f : A → B is: • injective or into if for every a1 , a2 ∈ A, f (a1 ) = f (a2 ) implies a1 = a2 ; • surjective or onto if rng(f ) = B ; or • bijective or one-to-one if it is both injective and surjective If f is injective then the inverse of f is the function f −1 : rng(f ) → dom(f ), defined by f −1 (f (a)) = a for every a ∈ dom(f ) Proposition 192 For every bijective function f : f −1 is a bijection; and (f −1 )−1 = f If f : A → B and g : B → C then the composition of f and g is the mapping gf : A → C defined by gf (a) = g (f (a)) for each a ∈ A