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Algebra II : Number Theory

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Example Question #5 : Number Sets

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:

$A\cup B\cup C$

Possible Answers:

$\left \{ 10, 20, 30, 40, 50 \right \}$

$\left \{ 40, 60 \right \}$

$\left \{ 20, 40, 60 \right \}$

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

$\left \{ 10, 20, 40, 60 \right \}$

Correct answer:

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

Explanation:

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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Example Question #6 : Number Sets

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:

$A\cap B$

Possible Answers:

$\left \{ 10, 20, 40 \right \}$

$\left \{ 20, 40, 60 \right \}$

$\left \{ 20, 40 \right \}$

$\left \{ 10, 20, 40, 60 \right \}$

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

Correct answer:

$\left \{ 10, 20, 40 \right \}$

Explanation:

The intersection is the set that contains the numbers found in both sets. Therefore the intersection is $\left \{ 10, 20, 40 \right \}$ .

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Example Question #1 : Number Sets

Sets3

Which set of numbers represents the union of E and F?

Possible Answers:

35,60,15,70,250

60,15,75,140,250,70

70,15

{35,70,15,60,75,250}

70,15,60,75

Correct answer:

{35,70,15,60,75,250}

Explanation:

The union is the set of numbers that lie in set E or in set F.

. Sets3

In this problem set E contains terms 35,60,15,70 , and set F contains terms 250,70,15,75 . Therefore, the union of these two sets is 35,60,15,70,75,250 .

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Example Question #8 : Number Sets

Express the following in Set Builder Notation:

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

Possible Answers:

$\left \{ x|-\infty < x< -3 \right \}$

$\left \{ x|-\infty< x< -3 \right\ OR\ -3< x< 1\ OR\ 1< x< \infty \}$

$\left \{ x|-3< x< \infty \right \}$

$\left \{ x|-\infty < x< 1 \right \}$

$\left \{ x|-3< x< 1 \right \}$

Correct answer:

$\left \{ x|-\infty< x< -3 \right\ OR\ -3< x< 1\ OR\ 1< x< \infty \}$

Explanation:

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

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

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Example Question #1 : Number Sets

Find the intersection of the two sets:

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

Possible Answers:

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

$\left \{ 1, 3, 5, 8, 9 \right \}$

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

$\left \{ \right \}$

Correct answer:

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

Explanation:

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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Example Question #1 : Number Sets

What type of numbers are contained in the set [3, 6, -4, 0, 8] ?

Possible Answers:

Integers

Natural

Complex

Irrational

Imaginary

Correct answer:

Integers

Explanation:

We can use process of elimination to find the correct answer.

It can't be Imaginary because we're not dividing by a negative number.

It can't be Complex because the number's aren't a mix of real and imaginary numbers.

It can't be Irrational because they aren't fractions.

It can't be Natural because there are negative numbers.

It must be Integers then! All the numbers are whole numbers that fit on the number line.

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Example Question #11 : Number Sets

True or false:

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

Possible Answers:

True

False

Correct answer:

False

Explanation:

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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Example Question #12 : Number Sets

What is $A\cap B$ ?

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

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

Possible Answers:

$\{9,11\}$

$\{1,11\}$

$\{1,9\}$

$\{1,9,11\}$

$\{1,2,3,4,5,6,8,9,11,15\}$

Correct answer:

$\{1,9,11\}$

Explanation:

$A\cap B$ 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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Example Question #12 : Number Sets

True or false:

The following set comprises only imaginary numbers:

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

Possible Answers:

False

True

Correct answer:

True

Explanation:

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.

Powers of i

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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Example Question #12 : How To Find A Solution Set

Set A is composed of all multiples of 4 that are that are less than the square of 7. Set B includes all multiples of 6 that are greater than 0. How many numbers are found in both set A and set B?

Possible Answers:

Correct answer:

Explanation:

Start by making a list of the multiples of 4 that are smaller than the square of 7. When 7 is squared, it equals 49; thus, we can compose the following list:

$\text{4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, and 48}$

Next, make a list of all the multiples of 6 that are greater than 0. Since we are looking for shared multiples, stop after 48 because numbers greater than 48 will not be included in set A. The biggest multiple of 4 smaller that is less than 49 is 48; therefore, do not calculate multiples of 6 greater than 48.

$\text{6, 12, 18, 24, 30, 36, 42, and 48}$

Finally, count the number of multiples found in both sets. Both sets include the following numbers:

$\text{12, 24, 36, and 48}$

The correct answer is 4 numbers.

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Algebra II : Number Theory

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #5 : Number Sets

Example Question #6 : Number Sets

Example Question #1 : Number Sets

Example Question #8 : Number Sets

Example Question #1 : Number Sets

Example Question #1 : Number Sets

Example Question #11 : Number Sets

Example Question #12 : Number Sets

Example Question #12 : Number Sets

Example Question #12 : How To Find A Solution Set

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