Number Sets - Algebra 2

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Question

True or false:

The following set comprises only imaginary numbers:

$\left \{ i^{7}, i^{9}, i^{11}, i^{13}, i^{15}\right \}$

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Answer

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Every element in the set $\left \{ i^{7}, i^{9}, i^{11}, i^{13}, i^{15}\right \}$ is equal to raised to an odd-numbered power, so when each exponent is divided by 4, the remainder will be either 1 or 3. Therefore, each element is equal to either or . Consequently, the set includes only imaginary numbers.

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Question

True or false:

The following set comprises only imaginary numbers:

$\left \{ i^{7}, i^{9}, i^{11}, i^{13}, i^{15}\right \}$

Tap to reveal answer

Answer

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Every element in the set $\left \{ i^{7}, i^{9}, i^{11}, i^{13}, i^{15}\right \}$ is equal to raised to an odd-numbered power, so when each exponent is divided by 4, the remainder will be either 1 or 3. Therefore, each element is equal to either or . Consequently, the set includes only imaginary numbers.

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Question

What is $A\cap B$ ?

$A=\{1,3,5,8,9,11,15\}$

$B=\{1,2,4,6,9,11\}$

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Answer

$A\cap B$ or A intersect B means what A and B have in common.

$A=\{{\color{Red} 1},3,5,8,{\color{Red} 9,11},15\}$

$B=\{{\color{Red} 1},2,4,6,{\color{Red} 9,11}\}$

In this case both A and B have the numbers 1, 9, and 11.

$A\cap B=\{1,9,11\}$

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Question

Find the intersection of the two sets:

$A=\left \{ 1, 2, 3, 4, 5, 6, 7, 8 \right \}, B=\left \{ 2, 4, 6, 7, 9 \right \}$

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Answer

To find the intersection of the two sets, $A\cap B$ , we must find the elements that are shared by both sets:

$\left \{ 2, 4, 6, 7 \right \}$

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Question

True or false:

The set $\left \{ i^{1}, i^{2}, i^{3}, i^{4}, i^{5}\right \}$ comprises only imaginary numbers.

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Answer

Any even power of the imaginary unit is a real number. For example,

$i^{2}= -1$ from the definition of as the principal square root of .

Also, from the Power of a Power Property,

$i^{4} =( i ^{2} )^{2} =(-1)^{2} = 1$

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Question

True or false:

The set $\left \{ i^{1}, i^{2}, i^{3}, i^{4}, i^{5}\right \}$ comprises only imaginary numbers.

Tap to reveal answer

Answer

Any even power of the imaginary unit is a real number. For example,

$i^{2}= -1$ from the definition of as the principal square root of .

Also, from the Power of a Power Property,

$i^{4} =( i ^{2} )^{2} =(-1)^{2} = 1$

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Question

True or false:

The set $\left \{ i^{1}, i^{2}, i^{3}, i^{4}, i^{5}\right \}$ comprises only imaginary numbers.

Tap to reveal answer

Answer

Any even power of the imaginary unit is a real number. For example,

$i^{2}= -1$ from the definition of as the principal square root of .

Also, from the Power of a Power Property,

$i^{4} =( i ^{2} )^{2} =(-1)^{2} = 1$

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Question

True or false:

The set $\left \{ i^{1}, i^{2}, i^{3}, i^{4}, i^{5}\right \}$ comprises only imaginary numbers.

Tap to reveal answer

Answer

Any even power of the imaginary unit is a real number. For example,

$i^{2}= -1$ from the definition of as the principal square root of .

Also, from the Power of a Power Property,

$i^{4} =( i ^{2} )^{2} =(-1)^{2} = 1$

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Question

If $A=\left \{ 10, 20, 30, 40, 50\right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60\right \}$ , find the following set:

$A\cup B\cup C$

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Answer

The union is the set that contains all of the numbers found in all three sets. Therefore the union is $\left \{ 10, 20, 30, 40, 50, 60 \right \}$ . You do not need to re-write the numbers that appear more than once.

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Question

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , find the following set:

$B\cap C$

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Answer

The intersection is the set that contains the numbers that appear in both $B=\left \{ 10, 20, 40, 60 \right \}$ and $C=\left \{ 20, 30, 40, 60 \right \}$ . Therefore the intersection is $\left \{ 20, 40, 60 \right \}$ .

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Question

What is $A\cap B$ ?

$A=\{1,3,5,8,9,11,15\}$

$B=\{1,2,4,6,9,11\}$

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Answer

$A\cap B$ or A intersect B means what A and B have in common.

$A=\{{\color{Red} 1},3,5,8,{\color{Red} 9,11},15\}$

$B=\{{\color{Red} 1},2,4,6,{\color{Red} 9,11}\}$

In this case both A and B have the numbers 1, 9, and 11.

$A\cap B=\{1,9,11\}$

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Question

If $A=\left \{ 10, 20, 30, 40, 50\right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60\right \}$ , find the following set:

$A\cup B\cup C$

Tap to reveal answer

Answer

The union is the set that contains all of the numbers found in all three sets. Therefore the union is $\left \{ 10, 20, 30, 40, 50, 60 \right \}$ . You do not need to re-write the numbers that appear more than once.

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Question

Express the following in Set Builder Notation:

$\left ( -\infty , -3\ \right )\cup \left ( -3, 1 \right )\cup \left ( 1, \infty \right )$

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Answer

$\left ( -\infty , -3 \right ) = \left \{x| -\infty < x< -3 \right \}$

and $\cup$ stands for OR in Set Builder Notation

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Question

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , find the following set:

$B\cap C$

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Answer

The intersection is the set that contains the numbers that appear in both $B=\left \{ 10, 20, 40, 60 \right \}$ and $C=\left \{ 20, 30, 40, 60 \right \}$ . Therefore the intersection is $\left \{ 20, 40, 60 \right \}$ .

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Question

Express the following in Set Builder Notation:

$\left ( -\infty , -3\ \right )\cup \left ( -3, 1 \right )\cup \left ( 1, \infty \right )$

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Answer

$\left ( -\infty , -3 \right ) = \left \{x| -\infty < x< -3 \right \}$

and $\cup$ stands for OR in Set Builder Notation

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Question

Express the following in Set Builder Notation:

$\left ( -\infty , -3\ \right )\cup \left ( -3, 1 \right )\cup \left ( 1, \infty \right )$

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Answer

$\left ( -\infty , -3 \right ) = \left \{x| -\infty < x< -3 \right \}$

and $\cup$ stands for OR in Set Builder Notation

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Question

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , then find the following set:

$A\cup B$

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Answer

The union is the set that contains all the numbers from $A= \left \{ 10, 20, 30, 40, 50 \right \}$ and $B=\left \{ 10, 20, 40, 60 \right \}$ . Therefore the union is $\left \{ 10, 20, 30, 40, 50, 60 \right \}$ .

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Question

Express the following in Set Builder Notation:

$\left ( -\infty , -3\ \right )\cup \left ( -3, 1 \right )\cup \left ( 1, \infty \right )$

Tap to reveal answer

Answer

$\left ( -\infty , -3 \right ) = \left \{x| -\infty < x< -3 \right \}$

and $\cup$ stands for OR in Set Builder Notation

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Question

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , find the following set:

$B\cap C$

Tap to reveal answer

Answer

The intersection is the set that contains the numbers that appear in both $B=\left \{ 10, 20, 40, 60 \right \}$ and $C=\left \{ 20, 30, 40, 60 \right \}$ . Therefore the intersection is $\left \{ 20, 40, 60 \right \}$ .

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Question

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , then find the following set:

$A\cup B$

Tap to reveal answer

Answer

The union is the set that contains all the numbers from $A= \left \{ 10, 20, 30, 40, 50 \right \}$ and $B=\left \{ 10, 20, 40, 60 \right \}$ . Therefore the union is $\left \{ 10, 20, 30, 40, 50, 60 \right \}$ .

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