Day 5 ­ Closure.notebook
September 11, 2012
Closure
SWBAT:
• Determine if a given set of numbers is closed under a given operation.
• Prove (support with an explanation) that a given set of numbers is or is not closed
under a given operation.
Do Now: Worksheet
HW#6--Closure Practice Worksheet
Quiz Thursday on BASIC Properties of operations NO CLOSURE / Sets
Test Friday 9/21
Sep 14­3:47 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Sep 13­8:11 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Distributive Property of Multiplication
over Addition
14
2x
Associative Property of Addition
5X
Commutative Property of Addition
0
1
Additive Inverse Property
Multiplicative Identity Property
0
Multiplicative Property of Zero
1
-1
Multiplicative Inverse Property
Multiplicative Property of -1
0
Additive Identity Property
3x
Distributive Property of Multiplication
over Subtraction
0
1
Opposite
Reciprocal
Sep 13­8:11 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Do Now Review:
Draw a graphic organizer to show the relationships among all the subsets of the set of
Real Numbers
Venn DiagramTree Diagram
True/False (1-6)/ Answer 7 and 8:
1. A whole number is an integer.5. If x is a positive integer then x≥0.
2. Zero is a positive integer.6. If x is a negative integer then x<0.
3. An integer is a whole number.7. What is the greatest negative integer?
4. One-half is not an integer8. What is the smallest positive integer?
Sep 14­3:50 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Property of Closure:
A set is closed under an operation if whenever the operation
(+, -, x, ÷) is performed within the set the answer is also a member
of the set.
When you are asked about closure you will be given:
a) A set of numbers
<AND>
b) An operation to be performed upon the set of numbers.
REMEMBER--Numbers can be OPERATED on with themselves.
Sep 14­3:52 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Example:
Given the set: {1, 2, 3 ...}
Is the set closed under:
Not Closed--show one "counter example"
Closed--show THREE cases that work.
Addition:
Subtraction:
Multiplication:
Division:
Sep 14­4:05 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Example:
Given the set: {0,1, 2, 3 ...}
Is the set closed under:
Not Closed--show one "counter example"
Closed--show THREE cases that work.
Addition:
Subtraction:
Multiplication:
Division:
Sep 14­4:16 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Example:
Given the set: {-1, 0, 1}
Is the set closed under:
Not Closed--show one "counter example"
Closed--show THREE cases that work.
Addition:
Subtraction:
Multiplication:
Division:
Sep 14­4:19 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Example:
Given the set: {Rational Numbers}
Is the set closed under:
Not Closed--show one "counter example"
Closed--show THREE cases that work.
Addition:Subtraction:
Multiplication:Division:
Sep 14­4:18 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Sep 14­4:29 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Sep 14­4:29 PM
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Day 5 ­ Closure.notebook
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Question: Is the set of negative integers closed under the operation of
SQUARING a number?
Why or Why not?
Sep 14­4:40 PM
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Day 5 ­ Closure.notebook
September 11, 2012
√
√
x
√
√
√
√
√
√
√
√
x
√
√
x
x
x
x
√
√
√
√
√
√
√
√
√
√
x
x
*
*
*
x
x
x
by zero is undefined. The set would be
*Division
closed under division if zero were excluded.
Sep 14­4:29 PM
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Day 5 ­ Closure.notebook
September 11, 2012
Sep 11­1:54 PM
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