unary operation An operation or a rule that applies
to single elements of a set to produce other (but not
necessarily different) elements of the set is called a
unary operation. For example, the rule that takes the
absolute value of a real number can be thought of as a
unary operation on the set of real numbers. The operation that reflects geometric shapes about the x-axis is a
unary operation on the set of all geometric figures in
the plane. However, the rule that associates each geometric figure with its area, for example, is not a unary
operation on this set, since the results are numbers, not
other geometric figures. A unary operation can be
thought of as a FUNCTION from a set to the same set.
See also BINARY OPERATION; OPERATION.
unique solution An equation is said to have a unique
solution if there is only one possible value of the variable(s) described in the equation that make the equation
true. For example, the equation x + 4 = 7 has the
unique solution x = 3, and the equation a
2 + b
2 = 0 has
the unique solution, in the realm of the real numbers,
(a,b) = (0,0). On the other hand, the equation x
2 – 9 = 0
has more than one solution (x could either be 3 or –3),
and the equation x
44 + 2y
6 = –7 has no real solutions.
Geometric arguments can sometimes be used to
determine whether or not a system of equations has a
unique solution. For example, the pair of equations
2x + 4y = 5
x + 3y = 8
represents two straight lines of different slopes. As two
nonparallel lines must intersect at a unique point,
there is a unique solution to the system. A system of
three equations representing three planes in space has
a unique solution only if those three planes meet at a
single point.
Sometimes a solution is called essentially unique if
the term uniqueness refers only to the underlying
structure of the problem and variations of the solution
do not affect mathematical content. For example, the
solution to the DIFFERENTIAL EQUATION
= x is
essentially unique. One solution is x
2 , and any other
solution to this problem differs from this one only by
the addition of a constant. In MODULAR ARITHMETIC,
x = 43 is, essentially, the unique solution to:
x ≡ 3 (mod 5)
x ≡ 7 (mod 12)
Any other solution to the problem differs from 43 by a
multiple of 60.
unit circle The circle of radius 1 centered at the origin is called the unit circle. In CARTESIAN COORDINATES
it has equation x
2 + y
2 = 1.
The unit circle plays a fundamental role in the
development of TRIGONOMETRY: the cosine and sine of
an angle θ are defined to be the x- and y-coordinates,
respectively, of a point on a unit circle located at an
1
–
2
dy –
dx
516
U
to single elements of a set to produce other (but not
necessarily different) elements of the set is called a
unary operation. For example, the rule that takes the
absolute value of a real number can be thought of as a
unary operation on the set of real numbers. The operation that reflects geometric shapes about the x-axis is a
unary operation on the set of all geometric figures in
the plane. However, the rule that associates each geometric figure with its area, for example, is not a unary
operation on this set, since the results are numbers, not
other geometric figures. A unary operation can be
thought of as a FUNCTION from a set to the same set.
See also BINARY OPERATION; OPERATION.
unique solution An equation is said to have a unique
solution if there is only one possible value of the variable(s) described in the equation that make the equation
true. For example, the equation x + 4 = 7 has the
unique solution x = 3, and the equation a
2 + b
2 = 0 has
the unique solution, in the realm of the real numbers,
(a,b) = (0,0). On the other hand, the equation x
2 – 9 = 0
has more than one solution (x could either be 3 or –3),
and the equation x
44 + 2y
6 = –7 has no real solutions.
Geometric arguments can sometimes be used to
determine whether or not a system of equations has a
unique solution. For example, the pair of equations
2x + 4y = 5
x + 3y = 8
represents two straight lines of different slopes. As two
nonparallel lines must intersect at a unique point,
there is a unique solution to the system. A system of
three equations representing three planes in space has
a unique solution only if those three planes meet at a
single point.
Sometimes a solution is called essentially unique if
the term uniqueness refers only to the underlying
structure of the problem and variations of the solution
do not affect mathematical content. For example, the
solution to the DIFFERENTIAL EQUATION
= x is
essentially unique. One solution is x
2 , and any other
solution to this problem differs from this one only by
the addition of a constant. In MODULAR ARITHMETIC,
x = 43 is, essentially, the unique solution to:
x ≡ 3 (mod 5)
x ≡ 7 (mod 12)
Any other solution to the problem differs from 43 by a
multiple of 60.
unit circle The circle of radius 1 centered at the origin is called the unit circle. In CARTESIAN COORDINATES
it has equation x
2 + y
2 = 1.
The unit circle plays a fundamental role in the
development of TRIGONOMETRY: the cosine and sine of
an angle θ are defined to be the x- and y-coordinates,
respectively, of a point on a unit circle located at an
1
–
2
dy –
dx
516
U