Number Sets - Algebra II

Card 0 of 14

Question

Express the following in Set Builder Notation:

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

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

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?

Answer

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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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:

$A\cup B$

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

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:

$B\cap C$

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 $C=\left { 20, 30, 40, 60\right }$ , find the following set:

$A\cap B\cap C$

Answer

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

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Question

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

$A\cap C$

Answer

The intersection is the set that contains the numbers found in both sets. Therefore the intersection is $\left { 20, 30, 40 \right }$ .

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Question

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

$A\cup B\cup C$

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:

$A\cap B$

Answer

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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Question

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

Answer

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

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In this problem set E contains terms , and set F contains terms . Therefore, the union of these two sets is .

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Question

Find the intersection of the two sets:

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

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

What is $A\cap B$ ?

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

What type of numbers are contained in the set ?

Answer

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 N atural 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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Question

True or false:

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

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 following set comprises only imaginary numbers:

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

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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