688
Modeling Arithmetic, Computation, and Languages
DefinitionS GrouP, CoMMutAtivE GrouP
3S, # 4 is a group if S is a nonempty set and # is a binary operation on S such that
1. # is associative.
2. an identity element exists (in S ).
3. each element in S has an inverse (in S ) with respect to # .
A group in which the operation # is commutative is called a commutative
group.
Once again, the dot in the definitions is a generic symbol representing a binary
operation. In any specific case, the particular binary operation has to be defined.
If the operation is addition, for example, then the + sign replaces the generic
symbol, as in 3Z, + 4. As an analogy with programming, we can think of the generic
symbol as a formal parameter to be replaced by an actual argument—the specific
operation—when its value becomes known. If it is clear what the binary operation
is, we may refer to “the group S” rather than “the group 3S, # 4.”
From the discussion, it should be clear that 3Z, + 4 is a commutative group,
with an identity element of 0. The idea of a group would not be useful if there
were not a number of other instances. (Again as an analogy with programming,
one can think of a group as an abstract data type—a pattern—with many possible
instances of that data type.)
example 1
Let R
+
denote the positive real numbers, and let # denote real-number multiplication, which is a binary operation on R
+
. Then 3R
+
, # 4 is a commutative group.
Multiplication is associative and commutative. The positive real number 1 serves
as an identity because
x # 1 = 1 # x = x
for every positive real number x. Every positive real number x has an inverse with
respect to multiplication, namely the positive real number 1∙x, because
x # 1∙x = 1∙x # x = 1
■
example 2
Let M 2 (Z) denote the set of 2 × 2 matrices with integer entries, and let + denote
matrix addition. Then + is a binary operation on M 2 (Z) (note that closure holds).
This is a commutative group because the integers are a commutative group, so
each corner of the matrix behaves properly. For example, matrix addition is commutative because
PrACtiCe 1 The set in Example 1 is limited to the positive real numbers. Is 3R, # 4 a commutative group?
Why or why not?
Modeling Arithmetic, Computation, and Languages
DefinitionS GrouP, CoMMutAtivE GrouP
3S, # 4 is a group if S is a nonempty set and # is a binary operation on S such that
1. # is associative.
2. an identity element exists (in S ).
3. each element in S has an inverse (in S ) with respect to # .
A group in which the operation # is commutative is called a commutative
group.
Once again, the dot in the definitions is a generic symbol representing a binary
operation. In any specific case, the particular binary operation has to be defined.
If the operation is addition, for example, then the + sign replaces the generic
symbol, as in 3Z, + 4. As an analogy with programming, we can think of the generic
symbol as a formal parameter to be replaced by an actual argument—the specific
operation—when its value becomes known. If it is clear what the binary operation
is, we may refer to “the group S” rather than “the group 3S, # 4.”
From the discussion, it should be clear that 3Z, + 4 is a commutative group,
with an identity element of 0. The idea of a group would not be useful if there
were not a number of other instances. (Again as an analogy with programming,
one can think of a group as an abstract data type—a pattern—with many possible
instances of that data type.)
example 1
Let R
+
denote the positive real numbers, and let # denote real-number multiplication, which is a binary operation on R
+
. Then 3R
+
, # 4 is a commutative group.
Multiplication is associative and commutative. The positive real number 1 serves
as an identity because
x # 1 = 1 # x = x
for every positive real number x. Every positive real number x has an inverse with
respect to multiplication, namely the positive real number 1∙x, because
x # 1∙x = 1∙x # x = 1
■
example 2
Let M 2 (Z) denote the set of 2 × 2 matrices with integer entries, and let + denote
matrix addition. Then + is a binary operation on M 2 (Z) (note that closure holds).
This is a commutative group because the integers are a commutative group, so
each corner of the matrix behaves properly. For example, matrix addition is commutative because
PrACtiCe 1 The set in Example 1 is limited to the positive real numbers. Is 3R, # 4 a commutative group?
Why or why not?