1. tables Objectives: By the end of class today I will:
Understand closure
Be able to read tables
Be able to find identies and inverses within a table

2. let's review our properties first1
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3. let's review our properties first3
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5. CLOSURE
When you combine any two elements of a set, the result is also included in the set.
Example: When you add two even numbers (from the set of even numbers), the sum is always even.
If an element outside the set is produced, then the operation is not closed. 
When you find ONE example that does not work, the set is not closed under that operation. 
Example: Even numbers are not closed under division because 100/4 = 25 and 25 is odd. 

6. CLOSURE The elements of this table are 1,2,3,4
Since the elements in this table are limited to 1,2,3,4 the table is CLOSED under the indicated operation

7. 1
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8. Are these statements True or False? If false, give a counterexample.
sage n scribe
Property:               Commutative
Are these statements True or False?
If false, give a counterexample.
Addition:                  
Subtraction:                  
Multiplication:              
Division:               

9. Are these statements True or False? If false, give a counterexample.
sage n scribe
Property:               Associative
Are these statements True or False?
If false, give a counterexample.
Addition:                                
Subtraction:                                
Multiplication:                        
Division:                          

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11. CLOSURE
The round smiley faces are a closed set.  No matter what operation is performed on round smiley faces, another round smiley face will be created.  Thus, there are always only round smiley faces in the box.

13. BINARY OPERATIONS
is simply a rule for combining two objects of a given type, to obtain another object of that type
Example: is a binary operation that you learned in elementary school
binary operations need not be applied only to numbers.
A binary operation on a finite set (a set with a limited number of elements) is often displayed in a table that demonstrates how the operation is performed.

14. BINARY OPERATIONS This table shows the operation * ("star").
The operation is working on the finite set A = { a, b, c, d }
Read the first value from the left hand column and the second value from the top row.  The answer is in the cell where the row and column intersect. 
Example, a * b = b,  b * b = c, c * d = b,  d * b = a

15. BINARY OPERATIONS – IDENTITY ELEMENT
What single element will always return the original value or
Where are all of the values in its row or column are the same as the row or column headings.
Examples: The identity element is a because    a * a = a,   b * a = b,   c * a = c,  d * a = d
And….

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20. inverse
What element, when paired with b, will return the identity element a?
The inverse element of b is d.      because   b * d = a OR
GO TO THE TABLE, AND CIRCLE EVERYWHERE THE IDENTITY APPEARS – (explain)

21. Is the operation * commutative?
a * b = b * a  is true since both sides equal b.
c * d = d * c  is true since both sides equal b.
Having to test ALL possible arangements could take forever!  There must be an easier way
Simply draw a diagonal line from upper left to lower right, and see if the table is symmetric about this line. If the table is symmetric, then the operation is commutative!

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24. Practice with Tables
Let’s practice