Math / Numbers & Operations

QuestionIndicate if the statement is true or false. If it is false, find a counterexample to prove the claim is false. (Recall that a set of number is closed under an operation if it will always produce another number in the same set.)
14. The set of irrational numbers is closed under addition.
15. The set of integers is closed under addition and multiplication.
16. The set of irrational numbers is closed under multiplication.

Studdy Solution

STEP 1

1. A set is closed under an operation if applying the operation to any elements of the set results in an element that is also in the set.
2. We need to determine whether the given statements about closure are true or false.
3. If a statement is false, we need to find a counterexample.

STEP 2

1. Evaluate closure of irrational numbers under addition.
2. Evaluate closure of integers under addition and multiplication.
3. Evaluate closure of irrational numbers under multiplication.

STEP 3

Evaluate the closure of the set of irrational numbers under addition.

STEP 4

Consider two irrational numbers: $\sqrt{2}$ and $-\sqrt{2}$ .

STEP 5

Add the two numbers: $\sqrt{2} + (-\sqrt{2}) = 0$ .

STEP 6

Since $0$ is a rational number, the set of irrational numbers is not closed under addition.
The statement "The set of irrational numbers is closed under addition" is FALSE.

STEP 7

Evaluate the closure of the set of integers under addition.

STEP 8

Consider two integers: $3$ and $5$ .

STEP 9

Add the two numbers: $3 + 5 = 8$ .

STEP 10

Since $8$ is an integer, the set of integers is closed under addition.

STEP 11

Evaluate the closure of the set of integers under multiplication.

STEP 12

Consider two integers: $3$ and $5$ .

STEP 13

Multiply the two numbers: $3 \times 5 = 15$ .

STEP 14

Since $15$ is an integer, the set of integers is closed under multiplication.
The statement "The set of integers is closed under addition and multiplication" is TRUE.

STEP 15

Evaluate the closure of the set of irrational numbers under multiplication.

STEP 16

Consider two irrational numbers: $\sqrt{2}$ and $\sqrt{2}$ .

STEP 17

Multiply the two numbers: $\sqrt{2} \times \sqrt{2} = 2$ .

STEP 18

Since $2$ is a rational number, the set of irrational numbers is not closed under multiplication.
The statement "The set of irrational numbers is closed under multiplication" is FALSE.

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Studdy Solution

STEP 1

1. A set is closed under an operation if applying the operation to any elements of the set results in an element that is also in the set.2. We need to determine whether the given statements about closure are true or false.3. If a statement is false, we need to find a counterexample.

STEP 2

1. Evaluate closure of irrational numbers under addition.2. Evaluate closure of integers under addition and multiplication.3. Evaluate closure of irrational numbers under multiplication.

STEP 3

Evaluate the closure of the set of irrational numbers under addition.

STEP 4

Consider two irrational numbers: 2\sqrt{2}2​ and −2-\sqrt{2}−2​.

STEP 5

Add the two numbers: 2+(−2)=0\sqrt{2} + (-\sqrt{2}) = 02​+(−2​)=0.

STEP 6

Since 000 is a rational number, the set of irrational numbers is not closed under addition. The statement "The set of irrational numbers is closed under addition" is FALSE.

STEP 7

Evaluate the closure of the set of integers under addition.

STEP 8

Consider two integers: 333 and 555.

STEP 9

Add the two numbers: 3+5=83 + 5 = 83+5=8.

STEP 10

Since 888 is an integer, the set of integers is closed under addition.

STEP 11

Evaluate the closure of the set of integers under multiplication.

STEP 12

Consider two integers: 333 and 555.

STEP 13

Multiply the two numbers: 3×5=153 \times 5 = 153×5=15.

STEP 14

Since 151515 is an integer, the set of integers is closed under multiplication. The statement "The set of integers is closed under addition and multiplication" is TRUE.

STEP 15

Evaluate the closure of the set of irrational numbers under multiplication.

STEP 16

Consider two irrational numbers: 2\sqrt{2}2​ and 2\sqrt{2}2​.

STEP 17

Multiply the two numbers: 2×2=2\sqrt{2} \times \sqrt{2} = 22​×2​=2.

STEP 18

Since 222 is a rational number, the set of irrational numbers is not closed under multiplication. The statement "The set of irrational numbers is closed under multiplication" is FALSE.

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1. A set is closed under an operation if applying the operation to any elements of the set results in an element that is also in the set.
2. We need to determine whether the given statements about closure are true or false.
3. If a statement is false, we need to find a counterexample.

1. Evaluate closure of irrational numbers under addition.
2. Evaluate closure of integers under addition and multiplication.
3. Evaluate closure of irrational numbers under multiplication.

Consider two irrational numbers: $\sqrt{2}$ and $-\sqrt{2}$ .

Add the two numbers: $\sqrt{2} + (-\sqrt{2}) = 0$ .

Since $0$ is a rational number, the set of irrational numbers is not closed under addition.
The statement "The set of irrational numbers is closed under addition" is FALSE.

Consider two integers: $3$ and $5$ .

Add the two numbers: $3 + 5 = 8$ .

Since $8$ is an integer, the set of integers is closed under addition.

Consider two integers: $3$ and $5$ .

Multiply the two numbers: $3 \times 5 = 15$ .

Since $15$ is an integer, the set of integers is closed under multiplication.
The statement "The set of integers is closed under addition and multiplication" is TRUE.

Consider two irrational numbers: $\sqrt{2}$ and $\sqrt{2}$ .

Multiply the two numbers: $\sqrt{2} \times \sqrt{2} = 2$ .

Since $2$ is a rational number, the set of irrational numbers is not closed under multiplication.
The statement "The set of irrational numbers is closed under multiplication" is FALSE.