Mathematical Systems
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Think of a switch for a fan where there are four settings. Off, High, Med, and Low. Use to represent turning the switch to the right (clockwise)
*This table has 4 elements {0,1,2,3}
*Binary Operation: a rule that can be used to combine any 2 elements and at least 1 binary operation
*Mathematical System: consists of a set of elements and at least one binary operation (ex ‐ the table)
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Example of a Mathematical System
Adding Even and Odd numbers:
Even + Even = Even + Odd =
Odd + Odd =
What is the set of elements of this mathematical system?
What is the binary operation of this mathematical system?
Use the table to find E+O. What does this mean?
Find O + O. What does this mean?
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5 Properties of Mathematical Systems
1. Closure Property
2. Commutative Property
3. Associative Property
4. Identity Property
5. Inverse Property
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The Closure Property
a mathematical system is CLOSED under binary operation if the answer to any possible combination of two elements of the set is an element that is in the set.
***IF THE RESULT IS AN ELEMENT OF THE SET, THEN THAT SET IS CLOSED (OR HAS CLOSURE) UNDER THE BINARY OPERATION***
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Example: Is the set {0,1,2,3} closed under the operation of addition?
+ 0 1 2 3 0 1
2
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Example: Consider the set of integers {...,‐3, ‐2, ‐1, 0 , 1, 2, 3, ...}
Is this set closed under the operation of multiplication?
Is the set of integers closed under the operation of division?
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Is the set of natural numbers, {0,1,2,3,4, ....} closed under the operation of multiplication?
Is the set of natural numbers closed under subtraction?
Is the set of natural numbers closed under the operation of division?
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HOMEWORK: pg 709 #1‐22
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