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Mathematical Systems
Study Guide
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Chapter 13Section 13.1
Hints:
- 1. When deciding if a set is closed under a particular operation,
always try to find one case where the answer to a problem which uses the
operation is not in the set. If the answer is not in the set, then the
set is not closed under that operation. For example, the set of
integers is not closed under division because when 1 is divided by 2, the
result is a fraction 1/2, which is not in the set of integers.
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2. For help in distinguishing the commutative property from the associative
property, try remembering the following.
Commutative comes from the word
"commute." For instance, if someone commutes to school, he
or she travels. In the commutative property, the terms can be moved or
"travel." -
- a
+ b = b + a
-
- On the left side of the equation above, a
is the first term and b is the second
term. On the right side of the equation, b
is the first term and a is the second term.
The terms have been rearranged or have "commuted."
Associative comes from the the word
"associate." An associate of a law firm belongs to the
group. In the associative property, the grouping of terms using
parentheses can be changed. -
- (a
+ b) + c = a + (b + c)
-
- On the left side of the equation above, a
and b are grouped together in
parentheses. On the right side of the equation, b
and c are grouped together in
parentheses. The terms have been regrouped, or they have change their
"association."
-
- 3. A person's identity is a reflection of who he or she is.
Think of an identity element in a mathematical system as any element that
yields a "reflection" of another element. For example, the
identity element of multiplication is 1. This is because 1 times a
number yields the number itself (its "reflection").
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Section 13.2
Hints:
- 1. The solution to a modular arithmetic problem , if one exists, will
always be a number from 0 through m-1. m is called the
modulus of the system. For example, in a modulo 6 system, the solution
will be a number from 0 through 5 because m=6.
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- 2. Use the following information when interpreting modulo system
notation.
-
- a = b(mod m)
-
- 1. a is the
number to be converted.
2. b is the
answer. 3. m is the
modulus of the system. -
- 3. Use the following information when working a modulo system
conversion.
-
- a = b(mod m)
-
- Step 1: Divide the
modulus, m,
into a.
Step 2: Write the
remainder from Step 1 in place of b. Step 3: Finish the
notation by placing (mod m)
next to b.
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