pqp∧q

TTT

TFF

FTF

FFF

94 connected

Conjunction circuit

A conjunction can be modeled via a series circuit. If T

denotes the flow of current, then current moves through

the circuit as a whole if, and only if, both switches p

and qallow the flow of current (that is, are closed).

See also

DISJUNCTION

.

connected Informally, a geometric object is “con-

nected” if it comes in one piece. For example, a region in

the plane is connected if, for any two points in that

region, one can draw a continuous line that connects the

two points and stays within the region. For example, a

HALF

-

PLANE

is connected. The set of all real numbers on

the number line different from zero is not connected.

A surface sitting in three-dimensional space, such

as a

SPHERE

or a

TORUS

, is connected if any two points

on the surface can be connected by a continuous path

that stays on the surface. A

GRAPH

is connected if, for

any two vertices, there is a contiguous path of edges

that connects them.

The connectivity of a geometric object is the num-

ber of cuts needed to break the shape into two pieces.

For example, a circle (interior and circumference) and a

solid sphere each have connectivity one. An

ANNULUS

and a torus each have connectivity two.

consistent A set of equations is said to be consistent

if there is a set of values that satisfies all the equations.

For example, the equations x+ y= 7 and x+ 2y= 11

are consistent, since they are satisfied by x= 3 and

y= 4. On the other hand, the equations x+ y= 1 and

x+ y= 2 are inconsistent.

In

FORMAL LOGIC

, a mathematical system is said to

be consistent if it is impossible to prove a statement to

be both true and not true at the same time. That is, a

system is consistent if it is free from

CONTRADICTION

.

Mathematicians have proved, for example, that arith-

metic is a consistent logical system.

See also

ARGUMENT

; G

ÖDEL

’

S INCOMPLETENESS

THEOREMS

;

LAWS OF THOUGHT

;

SIMULTANEOUS LINEAR

EQUATIONS

.

constant The word constant is used in a number of

mathematical contexts. In an algebraic expression, any

numeric value that appears in it is called a constant. For

example, in the equation y= 2x+ 5 with variables x

and y, the numbers 2 and 5 are constants. These num-

bers may be referred to as absolute constants because

their values never change. In general applications, how-

ever, constants may be considered to take any one of a

number of values. For example, in the general equation

of a line y= mx + bthe quantities mand bare consid-

ered constants even though they may adopt different

values for different specific applications.

A specific invariant quantity whose value is deter-

mined a priori, such as πor e, is also called a constant.

In physics, any physical quantity whose value is fixed

by the laws of nature, such as the speed of light c, or the

universal gravitational constant G, is called a constant.

The constant term in a

POLYNOMIAL

is the term

that does not involve any power of the variable. For

example, the polynomials x3– 5x+ 7 and 2z5– 3z2+ z

have constant terms 7 and 0, respectively.

A constant function is any function fthat yields the

same output value, asay, no matter which input value

is supplied: f(x) = afor all values x. The graph of a

constant function is a horizontal line. It is surprising,

for instance, to discover that the function given by

, defined for all positive numbers x, is a

constant function. The formula always returns the value

10 no matter which value for the input xis chosen.

constant of integration The

MEAN

-

VALUE THEO

-

REM

shows that any two antiderivatives of a given