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

k These properties of the powerset imply that, ifU is a universal set in which all sets of our interest are included, then∪, ∩, and are operations onP U.. Proposition 191 Properties of

k

These properties of the powerset imply that, ifU is a universal set in which all sets

of our interest are included, then∪, ∩, and  are operations onP( U) (We writeA for

U − A.) Thus, we obtain an algebraic structureP( U);∪ , ∩ ,  , ∅ , U  called the powerset

Boolean algebra ofU

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 setA to a set

Bis a rule denoted f :A → B which assigns to each element a ∈ A a unique element

f(a)∈ B The elementf(a) is called the value ofaunderf, or the image ofaunder

f Iff(a) =bthenais called a pre-image ofbunderf The set A is called the domain of f, denoted A=dom(f), andB is called the

co-domain , or the target set, off, denotedB =cod(f)

The notion of image can be generalized from elements to subsets of the domain as follows For any subsetX ⊆ Aof the domain off, the image ofXunderf is the set

f[X] ={ f(a)| a ∈ X }.

The image of the whole domainAunderf, that is, the setf[A] ={ f(a)| a ∈ A } of

all values off, is called the range or image off, also denotedrng(f)

Two functions f and g are equal iff dom(f) =dom(g), cod(f) =cod(g), and

f(a) =g(a) for everya ∈ dom( f)

The graph of a functionfis the set of ordered pairs{( a, f(a))| a ∈ dom( f)}.

A functionf :A → Bis:

• injective or into if for every a1, a2 ∈ A,f(a1) =f(a2) impliesa1 =a2;

• surjective or onto if rng( f) =B; or

• bijective or one-to-one if it is both injective and surjective.

Iff is injective then the inverse offis the functionf −1:rng(f)→ dom( f), defined

byf −1(f(a)) =afor everya ∈ dom( f)

Proposition 192 For every bijective function f :

1 f −1 is a bijection; and 2 (f −1)−1=f

Iff :A → B andg:B → C then the composition off andgis the mappinggf :

A → Cdefined bygf(a) =g(f(a)) for eacha ∈ A