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Closure
Closure
Lesson Presentation
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Holt
Algebra 1Algebra 1
Holt
McDougal
6-5-EXT Closure
Objective
Identify sets and the operations under
which they are closed.
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Holt McDougal Algebra 1
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Vocabulary
set
element
subset
closure
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Holt McDougal Algebra 1
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A set is a collection of objects. Each
object in a set is called an element of
the set. A set may have no elements, a
finite number of elements, or an infinite
number of elements. For example, N =
{1, 2, 3, …} describes the set of natural
numbers.
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A subset is a set contained entirely
within another set. For example,
A = {2, 6, 11, 50} is a subset of set N
above. Also, N is a subset of the set of
real numbers. The diagram below
shows other subsets of the real
numbers.
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Holt McDougal Algebra 1
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A set of numbers is closed, or has
closure, under a given operation if the
result of the operation on any two
numbers in the set is also in the set.
For example, the set of even numbers
is closed under addition, since the sum
of two even numbers is also an even
number.
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Holt McDougal Algebra 1
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Example 1:Determining Closure of Sets of Numbers
A. Is the set {–2, 0, 2} closed under
multiplication?
Multiple each pair of elements in the set. Check
whether each product is in the set
–2  –2 = 4 
–2  0 = 0 
–2  2 = –4 
00=0
02=4
22=4
The set {–2, 0, 2} is
not closed under
multiplication.
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Example 1 : Continued
B. Show that the set of irrational numbers is
not closed under division.
Find two irrational numbers whose quotient is
not an irrational number.
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1
2
1 is not irrational.
The set of irrational numbers is not closed under
division.
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Remember!
Remember the Commutative Property of Multiplication.
If 1 × (-1) = -1, then -1 × 1 = -1. Only one instance
needs to be tested.
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Check It Out! Example 1
Show that the set of whole numbers is not
closed under subtraction.
4 – 5 = –1
–1 is not a whole number
The set of whole numbers is not closed under subtraction.
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The sum of two rational numbers is rational.
2 4
+ = _________
3 5
-12 + 18 = _______
The product of two rational numbers is rational.
æ 2 öæ 4 ö
ç ÷ç ÷ = _________
è 3 øè 5 ø
( 25) (-8) = __________
(81 ) (27 ) = ____________
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The sum of two irrational numbers is
SOMETIMES irrational.
3 + 4 3 = ________
(
)
5+ 2 7 + -2 7 = ________
The product of two irrational numbers is
SOMETIMES irrational.
( 5) ( 6 ) = _________
( 7 ) ( 7 ) = __________
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Example Determine if the sum or product is a
rational or irrational number.
1.
4
( 81)
2. -8
3.
3 +8 3
2 +p
4. æ 2 ö
ç ÷ ( -6)
è 5ø
5.
9 2 +4 7
6.
9 + 64
1
2
1
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Holt McDougal Algebra 1
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Example 2: Determining Closure of Sets of
Polynomials
A. Is the set {0, 1, x, x + 1} closed under
addition? Explain.
x + (x + 1) = 2x + 1 
The set {0, 1, x, x + 1} is not closed under addition.
B. Show that the set of polynomials is closed under
multiplication.
Since each product of two polynomials will result in a
polynomial, the set is closed.
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Check It Out! Example 2
Is the set {x, x + 1, x2 – 1} closed under
division?
x+1 =1+ 1 
x
x
The set {x, x + 1, x2 – 1} is not closed under division.
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