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 DE MORGAN’S LAWS.
De Morgan’s laws If A and B are two subsets of a
universal set, and A′ and B′ denote their complements,
then the following two identities, known as De Morgan’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 VENN 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 A and B. Consequently, x belongs to at least one complement
A′ or B′, and so x ∈ A′∪B′.
Conversely, if y ∈ A′∪B′, then y does not
belong to one (or both) of A and B. Consequently, y is 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 intersection 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 A is interpreted as “the set of all instances
in which a claim p is true,” and B “the set of all
instances in which a claim q is true,” then De Morgan’s
laws in set theory translate to the following two identities in FORMAL LOGIC:
¬(p q) = (¬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 logician AUGUSTUS DE MORGAN (1806–71).
denumerable (enumerable, numerable) A COUNTABLE 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 diagonal 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 ℵ 0 . Every infinite set contains a
denumerable subset. This can be established as follows:
Suppose X is an infinite set. Let x 1 be any element of X. Since X is infinite, this is not the
only element of X. Let x 2 be another element
of X. Since X is infinite, these are not the only
two elements of X. Let x 3 be another element
of X. Thus continuing this way produces a list
of elements of X: x 1 , x 2 , x 3 , … This list represents a denumerable subset of X.
One can legitimately say, then, that a denumerable set
is the “smallest” type of infinite set. That is, ℵ 0 is 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
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 DE MORGAN’S LAWS.
De Morgan’s laws If A and B are two subsets of a
universal set, and A′ and B′ denote their complements,
then the following two identities, known as De Morgan’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 VENN 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 A and B. Consequently, x belongs to at least one complement
A′ or B′, and so x ∈ A′∪B′.
Conversely, if y ∈ A′∪B′, then y does not
belong to one (or both) of A and B. Consequently, y is 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 intersection 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 A is interpreted as “the set of all instances
in which a claim p is true,” and B “the set of all
instances in which a claim q is true,” then De Morgan’s
laws in set theory translate to the following two identities in FORMAL LOGIC:
¬(p q) = (¬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 logician AUGUSTUS DE MORGAN (1806–71).
denumerable (enumerable, numerable) A COUNTABLE 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 diagonal 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 ℵ 0 . Every infinite set contains a
denumerable subset. This can be established as follows:
Suppose X is an infinite set. Let x 1 be any element of X. Since X is infinite, this is not the
only element of X. Let x 2 be another element
of X. Since X is infinite, these are not the only
two elements of X. Let x 3 be another element
of X. Thus continuing this way produces a list
of elements of X: x 1 , x 2 , x 3 , … This list represents a denumerable subset of X.
One can legitimately say, then, that a denumerable set
is the “smallest” type of infinite set. That is, ℵ 0 is 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