Holt McDougal Algebra 1

6-5-EXT Closure6-5-EXT Closure

Holt Algebra 1

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Holt McDougal Algebra 1

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Holt McDougal Algebra 1

6-5-EXT Closure

Identify sets and the operations under which they are closed.

Objective

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Holt McDougal Algebra 1

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setelementsubsetclosure

Vocabulary

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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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Holt McDougal Algebra 1

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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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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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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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Holt McDougal Algebra 1

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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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2 1 is not irrational.

The set of irrational numbers is not closed under division.

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Holt McDougal Algebra 1

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Remember the Commutative Property of Multiplication. If 1 × (-1) = -1, then -1 × 1 = -1. Only one instance needs to be tested.

Remember!

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Holt McDougal Algebra 1

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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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Holt McDougal Algebra 1

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The sum of two rational numbers is rational.

-12 + 18 = _______

The product of two rational numbers is rational.

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Holt McDougal Algebra 1

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The sum of two irrational numbers is SOMETIMES irrational.

The product of two irrational numbers is SOMETIMES irrational.

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Holt McDougal Algebra 1

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Example Determine if the sum or product is a rational or irrational number.

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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 + 1x = 1 +

1x

The set {x, x + 1, x2 – 1} is not closed under division.

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