Concept Byte: Closure

Use With Lesson 1-6

ACTIVITY

What does it mean for a set of numbers to be closed under the operation of multiplication? Working with a partner, you will learn about closure.

Activity 1

1. Each person makes three cards as shown below labeled − 1 , negative 1 comma  0, and 1. On a turn, a player picks any two of the six cards and multiplies the numbers on the cards. The winner is the first person to find a product other than − 1 , negative 1 comma  0, or 1. Has anyone won the game after each person has taken 4 turns? After each person has taken 8 turns? Explain.

It is not possible to win the game in Exercise 1 because the product of any two numbers in the set is a number in the set. This means that the set { − 1 , negative 1 comma  0, 1} is closed under multiplication. When you perform an operation on any two numbers in a set and produce a number in the set, the set is closed under that operation. This property is called closure.

2. Repeat the game in Exercise 1 using addition instead of multiplication. Is the set closed under addition? Explain.
3. Determine whether it is possible to win each game. If so, give an example of a winning result.
   1. Add any two even numbers. Win by finding a result that is not an even number.
   2. Multiply any two negative numbers. Win by finding a product that is not a negative number.
   3. Add, subtract, or multiply any two integers. Win by finding a result that is not an integer.
4. Writing For each game in Exercise 3, determine whether the set of numbers is closed under the operation(s) used in the game. Explain your answers.

Activity 2

5. Determine whether it is possible to win each game. If so, give an example of a winning result.
   1. Find the absolute value of any integer. Win by finding a result that is not an integer.
   2. Square any negative number. Win by finding a result that is not a negative number.
   3. Square any rational number. Win by finding a result that is not a rational number.
6. Reasoning Under what arithmetic operation(s) does the set of whole numbers not have closure? Explain.

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