⊃ ⊃

⊃

162 empty set

of a general ellipsoid. This has proved to be a very dif-

ficult problem, and no closed-form formula exists. (The

surface area of an ellipse can only be expressed in

terms of a difficult “elliptic integral.”)

empty set (null set) Any set that contains no ele-

ments is called an empty set. For example, the set of all

real numbers greater than three and less than two is

empty, as is the set of all people with gills.

A set Ais said to be a subset of a set B, written

AB, if all elements of Abelong to B. Consequently

any empty set is a subset of any other set. In particular,

if Aand Bare both empty, then ABand BA,and

the two empty sets are equal. This shows that there is

only one empty set. It is usually denoted as Ø, but it

can also be written as { }.

The set with the empty set as its one member is

written {Ø}, and the set with the set containing the

empty set as its lone member is written {{Ø}}. In this

way we construct a chain of sets:

Ø, {Ø}, {{Ø}}, {{{Ø}}},…

which naturally corresponds to the sequence of count-

ing numbers 0, 1, 2, 3, … In this context one could

argue that all of mathematics arises from the empty set.

It is an interesting exercise then to give a numerical

interpretation to a two-member set of the form

{Ø,{Ø}}, for instance.

A set that is not empty is called nonempty.

See also

SET THEORY

.

endpoint See

INTERVAL

.

epicycle See

CYCLOID

.

epsilon-delta definition See

LIMIT

.

equality Two quantities are said to be equal if, in

some meaningful sense, they are equivalent. For exam-

ple, the quantities 2 + 3 and 5 have the same value and

so are equal. The two sets {a,b,c} and {c,a,b} are equal

since they contain the same elements. The symbol = is

used to denote the equivalence of two quantities, and

so we write 2+3 = 5 and {a,b,c} = {c,a,b}.

Two algebraic expressions are said to be equal if

one can be transformed into the other by the standard

rules of algebra. For instance, (x+ 1)2+ 3 = x2+ 2x+

4. Two functions are said to be equal if they have the

same domains and produce the same output value for

each input. For example, the functions f(x) = 9xand

, defined on positive values of x, are

equal.

The symbol = (a pair of parallel line segments to

denote equality) was introduced in 1557 by Welsh math-

ematician R

OBERT

R

ECORDE

(ca. 1510–58) “because

noe 2 thynges can be more equalle.”

See also

EQUATION

.

equating coefficients Two polynomials f(x) = anxn+

an–1xn–1 + … + a1x+ a0and g(x) = bnxn+ bn–1xn–1 + …

+ b1x+ b0are identical as functions, that is, give the

same output values for each input value of x, only if

the coefficients of the polynomials match: an=bn, an–1

= bn–1,…,a0= b0. (The general study of

POLYNOMIAL

s

establishes this.) The process of matching coefficients if

two polynomials are known to be the same is called

“equating coefficients.”

For example, if x2equals a polynomial of the form

A+ B(x– 1) + C(x– 1)(x– 2), then, after

EXPANDING

BRACKETS

, we have x2= Cx2+ (B– 3C)x+ (A– B+

2C). Equating coefficients yields: C= 1, B– 3C= 0

(and so B= 3), and A– B+ 2C= 0 (and so A= 1).

Thus x2= 1 + 3(x– 1) + (x– 1)(x– 2). (This technique

is often used in the method of

PARTIAL FRACTIONS

.)

As another example, if αand βare the roots of a

quadratic equation of the form x2– mx + n, then: x2–

mx + n= (x– α) (x– β) = x2– (α+ β)x+ αβ. We con-

clude then that mis the sum of the roots, and ntheir

product.

equating real and imaginary parts Two

COMPLEX

NUMBERS

a+ ib and c+ id are equal only if a= cand b

= d. Using this fact is called “equating real and imagi-

nary parts.” For example, if (x+ iy)(2 + 3i) = 4 + 5i,

then we must have 2x– 3y= 4 and 3x+ 2y= 5.

L

EONHARD

E

ULER

(1707–83) made clever use of

this technique to find formulae for P

YTHAGOREAN