Mathematical Systems and Groups

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Section 13.1
Math in Our World

• Mathematical Systems and Groups

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Learning Objectives

• Use an operation table to perform the operation
  in a mathematical system.
• Determine which properties of mathematical
  systems are satisfied by a given system.
• Decide if a mathematical system is a group.

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Mathematical Systems
A mathematical system consists of a finite or
infinite set of symbols and at least one
operation.
When youre driving on city streets and come to a
four-way intersection, you have four choices
right, left, straight, or U-turn.
In this case, the four symbols are R, L, S, and
U, representing your four choices at the
intersection. Consider what happens if you make
two of those choices consecutively.
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Mathematical Systems
Consider what happens if you make two of those
choices consecutively.
If you turn right and then left, you end up going
in the same direction you started in, so this
corresponds to going straight.
If you make a U-turn, then turn left, youll end
up in the same direction as if you had turned
right.
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Mathematical Systems
The symbols represent the combinations we already
mentioned R L S, and U L R.
The table represents a finite mathematical system
because there are a finite number of symbols with
an operation and is called an operation table.
A system with infinitely many symbols and an
operation is called an infinite mathematical
system.
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EXAMPLE 1 Using an Operation Table

• Use the table to find the result of each
  operation and describe what it means physically.
• (a) L S
• (b) R R
• (c) U S
• (d) (S U) R

R L S U
R U S R L
L S U L R
S R L S U
U L R U S
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EXAMPLE 1 Using an Operation Table
SOLUTION
(a) The symbol in row L and column S is L, so L
S L. If you turn left, then go straight, the
direction is the same as just turning left.
R L S U
R U S R L
L S U L R
S R L S U
U L R U S
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EXAMPLE 1 Using an Operation Table
SOLUTION
(b) The symbol in row R and column R is U, so
R R U. Turn right twice and your direction
is the same as if you had made a U-turn.
R L S U
R U S R L
L S U L R
S R L S U
U L R U S
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EXAMPLE 1 Using an Operation Table
SOLUTION
(c) U S U a U-turn followed by straight is
the same direction as the U-turn itself.
R L S U
R U S R L
L S U L R
S R L S U
U L R U S
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EXAMPLE 1 Using an Operation Table
SOLUTION
(d) (S U) R First, we find that S U U
then U R L. If you follow going straight
and making a U-turn with a right turn, its the
same direction as just turning left.
R L S U
R U S R L
L S U L R
S R L S U
U L R U S
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Five Properties of Mathematical Systems

• Closure Property
• For a system to be closed under an operation,
  when the operation is performed on any symbol,
  the result must be another symbol in the system.

The system of turns satisfies this property,
and is called a closed system because any
combination of two turns results in one of the
four basic turns.
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Five Properties of Mathematical Systems
2. Commutative Property A system is commutative
if the order in which you perform the operation
doesnt matter. More formally, a system is
commutative if for any two symbols a and b in a
system with some operation , a b b a.
The system of turns is commutative because the
result of any two consecutive turns is the same
regardless of the order you perform them in.
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Five Properties of Mathematical Systems
2. Commutative Property
R L S U
R U S R L
L S U L R
S R L S U
U L R U S
Notice that the table is symmetric about the
diagonal from upper left to lower right that is,
the bottom left and upper right triangular
portions are mirror images.
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Five Properties of Mathematical Systems
3. Associative Property A system has the
associative property if for any a, b, c in a
system with operation , (a b) c a (b
c).
There really isnt a quick way to show that a
system is associative you would have to try
every possible combination of three
symbols. Typically, well check a few examples to
try and get an idea of whether or not a system is
associative.
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Five Properties of Mathematical Systems
4. Identity Property A system has the identity
property if there is a symbol a in the system so
that a b b a b for any other symbol b.
In short, if theres a symbol that leaves all the
others unchanged when combined using the
operation, that symbol is called the identity
element for the system. For our system of
turns, you can see that S is the identity
element.
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Five Properties of Mathematical Systems
5. Inverse Property When a system has an identity
element, the next question to consider is whether
every symbol has another symbol that produces the
identity element under the operation.
That is, for any a in the system, is there always
an inverse element b so that a b is the
identity element? If there is, the system
satisfies the inverse property.
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EXAMPLE 2 Identifying the Properties of a
Finite Mathematical System

• Which properties does the system defined by the
  given table exhibit?

0 1 2 3
0 2 3 0 1
1 1 2 1 0
2 0 1 2 3
3 1 0 3 2
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EXAMPLE 2 Identifying the Properties of a
Finite Mathematical System
SOLUTION
0 1 2 3
0 2 3 0 1
1 1 2 1 0
2 0 1 2 3
3 1 0 3 2
Closure property Since every element in the body
of the table is in the set the system is defined
on, namely, 0, 1, 2, 3, the system is closed.
Commutative property Since the system is not
symmetric with respect to the diagonal, it does
not have the commutative property. For example, 0
1 3, but 1 0 1.
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EXAMPLE 2 Identifying the Properties of a
Finite Mathematical System
SOLUTION
0 1 2 3
0 2 3 0 1
1 1 2 1 0
2 0 1 2 3
3 1 0 3 2
Associative property Lets try some examples. (1
2) 3 1 3 0 1 (2 3) 1 3 0 This
one works! Lets try (1 0) 3 1 3 0 1
(0 3) 1 1 2
We found a counterexample, so the system is not
associative.
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EXAMPLE 2 Identifying the Properties of a
Finite Mathematical System
SOLUTION
0 1 2 3
0 2 3 0 1
1 1 2 1 0
2 0 1 2 3
3 1 0 3 2
Identity property There is an identity element,
2, since 2 x x 2 x for all x 0, 1,
2, 3
Inverse Property 0 and 0 are inverses, since 0
0 equals the identity.
1 and 1 are inverses, 2 and 2 are inverses, and 3
and 3 are inverses. Since every element has an
inverse, the system has the inverse property.
Therefore, the system exhibits closure, identity,
and inverse properties.
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Groups and Abelian Groups
A mathematical system is said to be a group if it
has closure, associative, identity, and inverse
properties.
A mathematical system is said to be an Abelian
group if, in addition to closure, associative,
identity, and inverse properties, it also has the
commutative property.
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EXAMPLE 3 Determining If a Mathematical System
is a Group

• Do the natural numbers under the operation of
  addition form a group? An Abelian group?

SOLUTION
The natural numbers are given by the set N 1,
2, 3, 4, .
Closure The natural numbers are closed under
addition that is, the sum of any two natural
numbers is a natural number. Therefore the system
is closed.
Associative The associative property of addition
holds for all real numbers, and since natural
numbers are real numbers, it holds for the
natural numbers as well. So the system is
associative.
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EXAMPLE 3 Determining If a Mathematical System
is a Group
SOLUTION
Identity There is no identity element in the set
of natural numbers (the identity element would be
0 under addition, but this element is not in N).
Inverse Since there is no identity element,
there can be no inverse. Therefore the inverse
property does not hold.
Since the system does not have the four
properties required to be a group, it is not a
group. Since it is not a group, it is also not an
Abelian group.
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EXAMPLE 4 Determining If a Mathematical System
is a Group

• Does the set - 1, 1 form a group under the
  operation of multiplication? An Abelian group?

SOLUTION
Closure The product of any two elements in the
set is also an element of the set, therefore the
system is closed.
Associative The associative property holds for
all elements in the set.
Identity The identity element is 1.
Inverse - 1 and - 1 are inverses as are 1 and 1.
Therefore each element has an inverse, so the
inverse property holds.
Since the system has all four properties, it is a
group. And, since the commutative property holds,
its also an Abelian group.