Objectives: By the end of class today I will:

Understand closureBe able to read tables

Be able to find identies and inverses within a table

If M and A represent integers, MAAM is an example of which property?

(1) commutative (3) distributive (2) associative (4) closure

Which expression is an example of the associative property? (1) (x + y) + z = x + (y + z) (2) x + y + z = z + y + x (3) x(y + z) = xy + xz (4) x • 1 = x

The equation *(∆ + ♥) = *∆ + *♥ is an example of the

(1) associative law (3) distributive law (2) commutative law (4) transitive law

Which equation illustrates the distributive property?

(1) 5(a + b) = 5a + 5b (3) a + (b + c) = (a + b) + c (2) a + b = b + a (4) a + 0 = a

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. 

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

Which set is closed under division?

(1) {1} (3) integers (2) counting numbers (4) whole numbers

Property:               Commutative

Are these statements

True or False?

If false, give a counterexample.

Addition:                   

 

Subtraction:                   

 

Multiplication:                 

Division:                   

   

Property:               Associative

Are these statements

True or False?

If false, give a counterexample.

Addition:                                    

Subtraction:                                    

Multiplication:                            

Division:                              

Ramón said that the set of integers is not closed for one of the basic operations (addition, subtraction, multiplication, or division).

You want to show Ramón that his statement is correct.

For the operation for which the set of integers is not closed, write an example using:

• a positive even integer and a zero

• a positive and a negative even integer

• two negative even integers

Be sure to explain why each of your examples illustrates that the set of integers is not closed for that operation.

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.

BINARY OPERATIONS

• is simply a rule for combining two objects of a given type, to obtain another object of that type

• Example: 2 + 4 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.

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

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….

What is the identity element for ♣ in the accompanying table?

(1) r (3) t (2) s (4) u

The operation element @ is determined by the following table:

What is the identity element of this operation?

(1) a, only (3) c (2) b, only (4) a and b

An addition table for a subset of real numbers is shown below. Which number is the identity element?

Explain your answer.

inverseWhat 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)

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!

In the addition table for a subset of real numbers shown below, which number is the inverse of 3?

Explain your answer.

The operation for the set {p,r,s,v} is defined in the accompanying table. What is the inverse element of r under the operation ?

(1) p (3) s (2) r (4) v

Practice with Tables

• Let’s practice