contrapositive 101

set of natural numbers, or can be put in one-

to-one correspondence with the entire set of

real numbers.

Despite his efforts, Cantor was unable to establish

whether or not his continuum hypothesis was true. In

1940 Austrian mathematician K

URT

G

ÖDEL

proved

that the continuum hypothesis cannot be proved false.

Unfortunately, as G

ÖDEL

’

S INCOMPLETENESS THEOREMS

show, this does not mean that the continuum hypothe-

sis is true: there exist statements in mathematics that

are undecidable, that is, ones that cannot be proved

true and cannot be proved false. It was suspected that

the continuum hypothesis might be such an undecid-

able statement. Twenty-three years later in 1963,

American logician Paul Cohen managed to prove that

this is indeed the case. Consequently one can either

deem the continuum hypothesis as true or as false, an

arbitrary choice, and be certain never to run into a

mathematical contradiction as a result.

contour integral (curvilinear integral, line integral)

If Cis a curve in the xy-plane and z= f(x,y) is a func-

tion of two variables, then one can attempt to compute

the surface

AREA

(one side) of a “wall” that follows the

curve C and has “height” the height of the function

above the curve. The integral that computes this, called

a contour integral and denoted ∫cf ds, is constructed by

selecting a large number of points p0, p1,…,pnalong

the curve Cand approximating the surface under con-

sideration by a collection of rectangular sections. The

ith rectangle can be taken to have base-length the dis-

tance between the points piand pi+1, which we denote

di, and height f(pi). The surface area is thus

approximated by the sum . Taking the limit

as we take finer and finer approximations defines the

desired contour integral.

If the curve Cis defined by parametric equations:

x= x(t) and y= y(t) for some parameter t, a ≤t ≤b,

then this procedure gives the contour integral as:

In physics and in advanced

VECTOR

calculus, one

also considers integrating, for example, the work done

in moving a particle along a curve Cthrough a

VECTOR

FIELD

(force field). Such considerations lead to other

types of integrals, also called line integrals.

contour line A line on a map that joins points of

equal height is called a contour line. Contour lines are

usually drawn for equal intervals of height. This gives

experienced map readers a clear mental picture of the

three-dimensional topography of the land: contour

lines close together, for example, indicate that the slope

of the land is steep.

In mathematics, contour lines are used to portray

the shapes of surfaces sitting in three-dimensional

space. For example, all points of the same height z= c

on the surface z= x2+ y2satisfy the equation x2+ y2=

cand so lie on a circle of radius √

–

c. This leads to a con-

tour map for the graph of the function f(x,y) = x2+ y2

consisting of sets of concentric circles about the origin.

The surface described is a

PARABOLOID

with vertex at

the origin.

contradiction In

FORMAL LOGIC

, any statement that

yields a

TRUTH TABLE

with final entries all false is called

a contradiction. For example, the statement (

¬

p) pis

a contradiction. From any contradiction, it is possible

to prove that any statement in mathematics is true. For

example, one can check that the compound statement

(

¬

p) p→qis a tautology. Consequently, in mathe-

matics, if one can prove that some statement pand its

negation

¬

pare both true, then since both (

¬

p) p

and (

¬

p) p→qare valid, no matter what statement

qrepresents, qis also true by inference. Any contradic-

tion that appears in mathematics would prove, for

example, that 1 = 2, and that every irrational number is

a fraction. Mathematicians sincerely hope that mathe-

matics is free from contradiction.

See also

CONSISTENT

.

contrapositive The contrapositive of a

CONDITIONAL

statement “p implies q” is the statement: “not q implies

not p.” It is the statement obtained by switching the

antecedent with the consequent, and negating each. For

example, the contrapositive of the statement, “If it is a

poodle, then it is a dog,” is “If it is not a dog, then it is

not a poodle.” The contrapositive of a statement is a

∨

∨

∨

∨