event, given that the number 1,849 is 43 squared. He

also observed that all those born in the year 1892

would enjoy a similar coincidence in the year 1936,

and those born in 1980 one in the year 2025. (The

number 2,025 is 45 squared.)

De Morgan died in London, England, on March

18, 1871.

See also D

E

M

ORGAN

’

SLAWS

.

De Morgan’s laws If Aand Bare two subsets of a

universal set, and A′and B′denote their complements,

then the following two identities, known as De Mor-

gan’s laws, hold:

(A∩B)′= A′∪B′

(A∪B)′ =A′∩B′

These identities can be used to convert any intersection

of sets into a union of sets, or vice versa.

These laws can be seen to hold true with the aid of

a V

ENN DIAGRAM

(by shading the region outside the

intersection or the union of the two sets), or by a formal

SET THEORY

argument. For instance, to prove the first

law, one must establish that any element that belongs to

(A∩B)′also belongs to A′∪B′, and vice versa. This can

be done as follows:

If x∈(A∩B)′, then x∉A∩B, meaning that x

does not belong to both Aand B. Conse-

quently, xbelongs to at least one complement

A′or B′, and so x∈A′∪B′.

Conversely, if y∈A′∪B′, then ydoes not

belong to one (or both) of Aand B. Conse-

quently, yis not an element of A∩B, and so

y ∈(A∩B)′.

The second law can be proved similarly.

De Morgan’s laws can be extended to the intersec-

tion or union of more than two sets. We have:

(A∩B∩C∩…∩Z)′= A∪B′∪C′∪…∪Z′

(A∪B∪C∪…∪Z)′= A′∩B′∩C′∩…∩Z′

When a set Ais interpreted as “the set of all instances

in which a claim pis true,” and B“the set of all

instances in which a claim qis true,” then De Morgan’s

laws in set theory translate to the following two identi-

ties in

FORMAL LOGIC

:

¬

(pq) = (

¬

p)∨(

¬

q)

¬

(p∨q) = (

¬

p)(

¬

q)

A

TRUTH TABLE

establishes that these corresponding

pairs of compound statements are logically equivalent.

These equivalences are also called De Morgan’s laws.

The formulae presented above were proposed in

1847 by Indian-born British mathematician and logi-

cian A

UGUSTUS

D

E

M

ORGAN

(1806–71).

denumerable (enumerable, numerable) A

COUNT

-

ABLE

infinite set is said to be denumerable. Thus a

denumerable set is any infinite set whose elements can

be placed in a list akin to the list of natural numbers 1,

2, 3, … The first

DIAGONAL ARGUMENT

shows that the

set of

RATIONAL NUMBERS

is denumerable. The diago-

nal argument of the second kind establishes that the set

of

REAL NUMBERS

is not. In some definite sense then,

the set of real numbers is a “larger” infinite set than

the infinite set of rationals. A denumerable set is said to

have

CARDINALITY

ℵ0. Every infinite set contains a

denumerable subset. This can be established as follows:

Suppose Xis an infinite set. Let x1be any ele-

ment of X. Since Xis infinite, this is not the

only element of X. Let x2be another element

of X. Since Xis infinite, these are not the only

two elements of X. Let x3be another element

of X. Thus continuing this way produces a list

of elements of X: x1, x2, x3, … This list repre-

sents a denumerable subset of X.

One can legitimately say, then, that a denumerable set

is the “smallest” type of infinite set. That is, ℵ0is the

“smallest” transfinite cardinal number.

See also

CONTINUUM HYPOTHESIS

.

derivative See

DIFFERENTIAL CALCULUS

.

Desargues, Girard (1591–1661) French Geometry,

Engineering Born on February 21, 1591, in Lyon,

France, mathematician Girard Desargues is considered

the founder of

PROJECTIVE GEOMETRY

, an innovative,

non-Greek, approach to geometry. His highly original

and famous 1639 text Brouillon project d’une atteinte

aux evenemens des recontres du cone avec un plan

∨

∨

Desargues, Girard 123