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 Mathematical Systems Study Guide

Chapter 13

Section 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.

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").

 Additional Exercises: 1. Q:  Use the table shown to explain whether or not the mathematical system is closed under the binary operation *. 2. Q: Consider the set of integers.  What is the inverse of -5 under the operation of addition? Click Here for Answer 3. Q: Is the set of even natural numbers closed under addition? Click Here for Answer

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.

2. Use the following information when interpreting modulo system notation.

a = b(mod m)

1.  a is the number to be converted.

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.

 Additional Exercises: 1. Q:  Determine if the statement is true or false. 47 = 2(mod 6)Click Here for Answer 2. Q:  Find the sum in the following. (12+3)mod 14 3. Q: Suppose that today is Tuesday, September 18.  Which day of the week will it be 72 days from now? Click Here for Answer

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