Boolean Algebra

Another example of an algebra is provided by the operations such as the

intersection

and the

union

of subsets S and

T

of a given set X. If we write X E S for "x is an element of S" and � for "if and only if', these operations are specified by giving the elements of the resulting subset of X as follows :

9. Boolean Algebra

Intersection Union

X E S n

T

� X E S and

x

E

T,

X E S U

T

^� X E S or

x

E

T,

27

( 1 )

(2)

=> X E S =>

T

� if X E S , then

x

E

T,

(3)

�

x

E

T,

or not

(x

E S) . They correspond exactly to the three propositional connectives "and",

"or", and "if then". They may also be pictured by Venn diagrams; if the set

X

is taken to be all the points in a rectangle while S and

T

are respec­

tively the points inside the ovals S and

T,

then two of these operations may be indicated by shaded areas as in Figure

1 .

There is also a unary operation, the

complement

.., S of S:

x

E .., S � not

(x

E S) (4)

These various operations n, U, => , .., satisfy certain algebraic identities which can all be deduced from a suitable list of axioms, the axioms for

Boolean Algebra.

Thus the set

P(X)

of all subsets of

X

is a Boolean alge­

bra.

There also are operations on infinite families of sets. Thus if S; is a sub­

set of

X

for each

i

in some "index" set

I,

the (infinite) Union and intersec­

tion are defined by

x

E U ^{S; � For some}

i

in

I,

X E S; ,

;

x

E n S; � For every

i

in

I,

X E S;

;

(5)

(6) These operations correspond to the logical quantifiers "There exists an

i"

and "For all

i",

respectively. These connections with logic will be explored further in Chapter XI.

Boolean algebra provides a Mathematical way of representing proper­

ties, in that each property H of elements of a set

X

determines a subset of

X;

namely, the subset S consisting of all those elements which have the property

S =

{x I x

E

X

and

x

has H} .

s

m

^T

s n T S � T

Figure I . Boolean operations.

(7)

2

8

I . Origins of Formal Structu re This subset is sometimes called the

extension

of the property H, to emphasize the notion that differently formulated properties may have the same extension-and that Mathematics has to do with extensions rather than with meanings. This in turn involves the "extensionality" axiom for sets-that a set is completely determined just by specifying its elements.

This means that the equality of two subsets of

X

is described by the state­

ment

S = T � (For all

x

in

X,

X E S �

x

E

n , (8)

while the inclusion of one subset S in another is described by

S c T � (For all

x

in

X,

X E S �

x

E n ; (9) here the arrow � stands for "implies".

This inclusion relation is transitive, reflexive, and antisymmetric, as these properties were defined in §4 above. In general, an

ordered set W

is a set

W

(such as

P(X»

with a binary relation (such as S c T for S, T E W) which is transitive, reflexive, and antisymmetric. An ordered set is often said to be

partially

ordered (a

poset)

because it need not satisfy the "trichotomy" property which holds for a linear order.

It is important to recognize that many orders are just partial orders and not total orders (i.e.,

not

linear). However, in many domains of the appli­

cation of Mathematics to social phenomena, there is a strong tendency to order ideas, people, and institutions in a

linear

way-for example, accord­

ing to rank on some imagined numerical measure. The more relevant notion of partial order seems little known and less used.

Diagrammatic presentation of an inclusion relation is suggestive. Thus the various inclusions of the subsets of a three-element set can be pictured by the rising lines in Figure 2, where the bottom symbol 0 denotes the

empty

subset. The Boolean operations on subsets may be visualized in this figure. For example, the union { l ,2 } of the subsets { l } and { 2 } is the smallest subset which lies "above" both the subsets { l } and {2 } ; in this way it is the least upper bound, as defined in §4, of { l } and { 2 } . Gen­

erally, the union S U T of two subsets S and T of a set

X

has the proper­

ties

S c S u T, T C S U T, S c R and T c R � S U T c R ,

( 1 0) ( I I ) which state that i t i s the least upper bound o f S and T in the partial order given by inclusion. In an exactly dual way, the intersection S n T is the greatest lower bound of the subsets S and T. In other words, both these Boolean operations can be described directly in terms of inclusion,

1 0. Calculus, Continuity, and Topology

(1,2,3)

--- I ^�

(I ,

2} (1, 3}

(2'13}

---��

Figure 2. Lattice of subsets.

29

without any use of membership. In Chapter XI we will see further exam­

ples of sets treated without the use of elements.

There are corresponding definitions for other inclusion relations. In general (and in view of diagrams like that above) a poset is said to be a

lattice

when it has a top element

I ,

a bottom element 0 and when each pair of elements have a least upper bound (called their

join)

and a greatest lower bound (called their

meet).

The lattice of subobjects of an algebraic object is a way of describing some of the structure of that object.

1 0. Calculus, Continuity, and Topology

Many notions besides those of transformation groups arise from the mathematical analysis of motion. The complex motions of the planets and the varying velocities of falling bodies suggest the idea of "rate of change" : Velocity as rate of change of distance or acceleration as rate of change of velocity. These ideas were codified in the notion of the deriva­

tive, subsequently formalized (Chapter VI) in the rigorous foundation of the calculus, as based on the axioms for the real numbers. This uses the definition of the derivative by means of limits and thus the consideration of a class of "good" functions-those which are differentiable. As a first example of this circle of ideas, we examine here another good class-the functions which are continuous.

A rigid motion M:

F

^-+

F

of a figure is continuous because (by rigidity) the distance from Mp to Mq must equal that from p to q. For a function f: R ^-+ R on the real numbers R continuity means considerably less: Just that

fx

and fy will be close if the originals

x

and y are sufficiently close.

This formulation is still pretty vague ; it should mean that one can make

fx

and fy "as close as you please" by requiring

x

to be "suitably close"

to y. This is still vague. "As close as you please" should mean "within a specified measure 8 (a positive real number) of closeness; "suitably close"

should mean that one can specify a measure of closeness (again a positive real number t:) which will do the job. All this (and we have telescoped a

30 I . Origins of Formal Structure

long and painful historical development) comes down to make the famil­

iar (but meticulous) £ - 8 definition of continuity: A function

f:

R --> R is

continuous

at a point

a

E R if

For all real £ > 0 there is a real 8 > 0 such that, for all

x

in R ,

( I )

If

I x

-

a I

< 8 , then

I f(x ) - f(a) I

< (

.

(2) If this statement holds for

all

points

a

E R, the function

f

is called con­

tinuous; the class of all such continuous functions is called

C.

Note that the statement involves both propositional connectives ("if . . . then") and the so called "bounded" quantifiers (For all real numbers, there exists a real number). Thus it is that careful formulations lead to the use of concepts of formal logic.

Topological and metric spaces arise from analysis of this definition of continuity. The inequalities used in the definition arise from ideas of approximation (approximations of the value b =

f(a )

to within the accu­

racy £ ) and so implicitly involve the open interval I. ( b ) =

{y I I y

- b

I

< £ } of center b and "radius" £. In the familiar represen­

tation of the function

f

by its

graph

(the set of points

(x,

f(x » in the plane), this open interval appears as an open horizontal strip of width 2£

around

y

=

f(a )

(Figure 1 ). The definition is concerned with those points

x

E R for which f(x ) lands in this interval I = I. (b )-this set of points is usually called the

inverse image

of I under the function

f,

in symbols :

f- II =

{x I x

E R

and f(x )

E I}

.

Indeed, if Xo E

f

^-II (that is, if

f

(

x

o) E I), then one can prove from the definition of continuity that there is an open interval (on the

x

axis) of center Xo wholly contained in f - II. This amounts to the

Theorem.

The function f:

R --> R

is continuous for all a

E R

if and only if the inverse image f

^-I I

of every open interval of

R

is a union of open intervals.

- - - - - t

f(a) ⁼ b 2€

Figure I

1 0. Calculus, Continuity, and Topology 3 1

In other words, continuity can be described wholly in terms of open intervals.

Continuity is also needed for functions

f

defined on only a part of the real line, for functions of two or more variables, or for functions defined on a surface, etc. But this new definition requires no new ideas. The absolute value

I x - a I

which appears in the definition

( I )

and (2) is just the distance

p(x,a)

on the line R from the point

x

to the point

a.

Hence the same definition will work with any suitable notion of distance-that is, for a metric space.

Definition. If

X

and Y are metric spaces, a function

f: X

... Y is continu­

ous at a point

a

E

X

if for all real t: > 0 there is a real 8 > 0 such that for all

x

in

X,

if

p(x,a)

< 8 then

p(f(x),f(a»

< t:.

In particular, this defines continuity for functions of two real numbers

x,y.

Just regard

(x,y)

as the (coordinates of) a point in the plane R X R with the usual (pythagorean) metric P I =

p,

p«x,y),(a,b »

=

[(x - a i

+

(y - b i ll/2 ,

Then in the definition of continuity the open interval consisting of all

x

with

I x - a I

< 8 is replaced by the

open disc

with center

(a,b )

and radius 8. This disc consists of all points

(x,y)

in the plane with

(3) As in the theorem above, one can prove that a function

f:

R X R ... R of two real variables is continuous if and only if the inverse image of each open interval in R is a union of open discs in R X R.

But the plane is also a metric space with a distance

P2

described as the

"shortest path along the rectangular grid", so that

P2«x,y), (a,b»

=

I x - a I

+

I y - b I '

or with the distance given by

P3«X,y),

(a, b»

= Max(

I x - a I , I y - b I ) .

For a function

f(x,y)

of two real variables, any one of these distance for­

mulas may be used to define the continuity of

f

in the usual t: - 8 style.

These different distances all yield the

same

continuous functions, so that continuity, viewed invariantly, depends not on distance but on something else more intrinsic. What is it? The answer is well-known. Each distance Ph f>2, or P3 will determine a series of "open circular discs" -circular in that metric-of the respective forms (interior of) circle, diamond, or square (Figure 2). Within each disc we can readily draw a smaller disc of the

32 I. Origins of Formal Structure

Figure 2

other forms with the same center-so we get the same ( - 8 continuous function by just choosing different 8's to a given (.

What is the intrinsic formulation? In the alternative description of con­

tinuity stated in the theorem above there appeared unions of open inter­

vals and unions of discs. In any metric space

X

the open

disc

with center

a

and radius 8 may be defined to be the set of all points x in

X

with

p(x^,

a

) < 8. Now define an

open set U

in

X

to be any union (finite or infinite) of open discs. Equivalently, a subset

U

is

open

in

X

if to each point

a

E

U

there is a 8 > 0 such that p(x^,

a

) < 8 implies x E

U

-every point

a

in

U

is contained in an open disc centered at

a

and within

U.

Now the continuity of f:

X

^... Y is expressed in terms of open sets: f is continuous if and only if the inverse image of any open set of Y is an open set of

X.

This is the desired intrinsic formulation, independent of the perhaps accidental choice of a metric. Specifically, the three different metrics p = P I , P 2 , and ^P3 described above for the plane all yield the same open sets, because any union of circular open discs is also a union of open squares or of open diamonds, and conversely. In this way, the notion of "open set" is more intrinsic than that of distance.

This suggests that a space can be defined directly in terms of its open subsets. A

topological space

is a set

X

in which certain of the subsets

U

are distinguished and called the

open

sets, and in which these open sets are required to satisfy the following three axioms :

1 . The intersection of two open sets is open;

2. The union of any collection of open sets is open;

3.

X

itself and the empty subset 0 c

X

are open.

A

topology

on a set

X

is then the specification of open subsets which satisfy these axioms. Thus any metric

X

determines a topology, in which the open sets are the unions of discs open in the metric. There are also topologies not defined by way of a metric. For example, there is a topol­

ogy on the set

N

of natural numbers for which the open subsets are the empty subset and all those subsets S with finite complement (in

N).

There are many other examples of topologies.

The definition of continuity (the inverse image of any open set is open) now applies to any function f:

X

^... Y between topological spaces

1 0. Calculus, Continuity, and Topology 33

Extensive experience has shown that this description of a "topology" in terms of open sets and neighborhoods is extraordinarily effective in for­

mulating all sorts of Mathematical facts in a geometric form. The concept of "topology" has been appropriately abstracted from the many examples of "continuity".

The notion of a topological space was first presented by F. Hausdorff in a famous (and beautiful) book

Mengenlehre.

His definition was formu­

lated differently, in terms of selected neighborhoods, and included an added axiom (the Hausdorff separation axiom): Two distinct points have disjoint neighborhoods. A topological space with this property is called a

In document Mathematics Form and Function (Page 37-44)