MATH 110 Sec 2-2: Comparing Sets Practice Exercises

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember that two sets are equal if and only if they have exactly the same elements.

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

is the empty set. That means that it does not have ANY elements.

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

is the empty set. That means that it does not have ANY elements. is also the empty set.

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

is the empty set. That means that it does not have ANY elements. is also the empty set. is no longer empty.

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

is the empty set. That means that it does not have ANY elements. is also the empty set. is no longer empty.

An empty set, , can’t be equal a set that is not empty, .

True or FalseThe sets and { } are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

is the empty set. That means that it does not have ANY elements. is also the empty set. is no longer empty.

An empty set, , can’t be equal a set that is not empty, .

The sets are NOT equal and the statement above is FALSE.

True or FalseThe sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

True or FalseThe sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember that two sets are equal if and only if they have exactly the same elements.

True or FalseThe sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

It is easy to see here that these two sets do not have exactly the same elements.

Remember that two sets are equal if and only if they have exactly the same elements.

True or FalseThe sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

It is easy to see here that these two sets do not have exactly the same elements.

Remember that two sets are equal if and only if they have exactly the same elements.

The sets are NOT equal and the statement above is FALSE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

{ 12 , 82 , 99 } and { a , e, p } are equivalent.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

{ 12 , 82 , 99 } and { a , e, p } are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

{ 12 , 82 , 99 } and { a , e, p } are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

321321

There are 3 elements in each set so the two sets are equivalent.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

{ 12 , 82 , 99 } and { a , e, p } are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

321321

There are 3 elements in each set so the two sets are equivalent.

The statement above is TRUE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equivalent.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

321 4 5 321 4 5

There are 5 elements in each set so the two sets are equivalent.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o , u} are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

321 4 5 321 4 5

There are 5 elements in each set so the two sets are equivalent.

The statement above is TRUE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o} are equivalent.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o} are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o} are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

321 4 5 321 4

There are 5 elements in one set and 4 in the other setso the two sets are NOT equivalent.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

The sets {1, 2 , 3 , 4 , 5} and {a , e , i , o} are equivalent.

Remember that two sets are equivalent if they have the same number of elements.

321 4 5 321 4

There are 5 elements in one set and 4 in the other setso the two sets are NOT equivalent.

The statement above is FALSE.

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember, the set of integers is the set of counting numbers(the positive integers) plus the set of negative integers plus zero.

{ … ,-3 , -2 , -1 , 0 , 1 , 2 , 3 , … }

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember, the set of integers is the set of counting numbers(the positive integers) plus the set of negative integers plus zero.

{ … ,-3 , -2 , -1 , 0 , 1 , 2 , 3 , … }

True or False (Justify your answer.) and are equal.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember, the set of integers is the set of counting numbers(the positive integers) plus the set of negative integers plus zero.

{ … ,-3 , -2 , -1 , 0 , 1 , 2 , 3 , … }

This statement is TRUE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

And we learned that the empty set () is a subset of EVERY set.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

So this statement is TRUE.

And we learned that the empty set () is a subset of EVERY set.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

99}

And we learned that the empty set () is a subset of EVERY set.

So this statement is TRUE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

99}Set A is a proper subset of set B (written A B)

if every element of A is also an element of B but A ≠ B.

And we learned that the empty set () is a subset of EVERY set.

So this statement is TRUE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

And we learned that the empty set () is a subset of EVERY set.

99}Set A is a proper subset of set B (written A B)

if every element of A is also an element of B but A ≠ B.We also learned that the empty set () is a proper subset of EVERY set.

So this statement is TRUE.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises True or False

99}Set A is a subset of set B (written A B)

if every element of A is also an element of B.

And we learned that the empty set () is a subset of EVERY set.

99}Set A is a proper subset of set B (written A B)

if every element of A is also an element of B but A ≠ B.We also learned that the empty set () is a proper subset of EVERY set.

This statement is also TRUE.

So this statement is TRUE.

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

321 4 5 76

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

321 4 5 76

So A has subsets

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

321 4 5 76

So A has subsets

How many proper subsets does A have?

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

321 4 5 76

So A has subsets

How many proper subsets does A have?A proper subset of set A does not include A itself.

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

321 4 5 76

So A has subsets

How many proper subsets does A have?

Therefore A has one less proper subset than

A proper subset of set A does not include A itself.

A = {11 , 12 , 13 , 14 , 15 , 17 , 18}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

How many subsets does A have?A set with k elements has subsets.

321 4 5 76

So A has subsets

How many proper subsets does A have?

So A has proper subsetsTherefore A has one less proper subset than

A proper subset of set A does not include A itself.

Use the table to find the number of subsets of the set of students who are either freshmen or athletes, or both.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

MAJOR CLASS RANK GPA ACTIVITIES

Gina History Freshman 3.8 Band

Dana Biology Freshman 1.4 Yearbook

Elston Business Freshman 1.7 Baseball

Frank French Senior 1.6 Soccer

Brenda History Junior 3.1 Tennis

Carmen Business Senior 3.7 Basketball

Use the table to find the number of subsets of the set of students who are either freshmen or athletes, or both.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

MAJOR CLASS RANK GPA ACTIVITIES

Gina History Freshman 3.8 Band

Dana Biology Freshman 1.4 Yearbook

Elston Business Freshman 1.7 Baseball

Frank French Senior 1.6 Soccer

Brenda History Junior 3.1 Tennis

Carmen Business Senior 3.7 Basketball

Use the table to find the number of subsets of the set of students who are either freshmen or athletes, or both.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

MAJOR CLASS RANK GPA ACTIVITIES

Gina History Freshman 3.8 Band

Dana Biology Freshman 1.4 Yearbook

Elston Business Freshman 1.7 Baseball

Frank French Senior 1.6 Soccer

Brenda History Junior 3.1 Tennis

Carmen Business Senior 3.7 Basketball

Use the table to find the number of subsets of the set of students who are either freshmen or athletes, or both.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

MAJOR CLASS RANK GPA ACTIVITIES

Gina History Freshman 3.8 Band

Dana Biology Freshman 1.4 Yearbook

Elston Business Freshman 1.7 Baseball

Frank French Senior 1.6 Soccer

Brenda History Junior 3.1 Tennis

Carmen Business Senior 3.7 Basketball

Use the table to find the number of subsets of the set of students who are either freshmen or athletes, or both.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

MAJOR CLASS RANK GPA ACTIVITIES

Gina History Freshman 3.8 Band

Dana Biology Freshman 1.4 Yearbook

Elston Business Freshman 1.7 Baseball

Frank French Senior 1.6 Soccer

Brenda History Junior 3.1 Tennis

Carmen Business Senior 3.7 Basketball

So, there are 6 students who

are either freshmen or

athletes (or both)

Use the table to find the number of subsets of the set of students who are either freshmen or athletes, or both.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

MAJOR CLASS RANK GPA ACTIVITIES

Gina History Freshman 3.8 Band

Dana Biology Freshman 1.4 Yearbook

Elston Business Freshman 1.7 Baseball

Frank French Senior 1.6 Soccer

Brenda History Junior 3.1 Tennis

Carmen Business Senior 3.7 Basketball

So, there are 6 students who

are either freshmen or

athletes (or both)

A set with 6 elements has subsets

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

List all the subsets of the set given below.A = {blackberry , blueberry , lemon}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Remember: A set with k elements has subsets.

So here, there are subsets.

321

Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø{blackberry}{blueberry}

{lemon}

{blackberry, blueberry}{blackberry, lemon}{blueberry, lemon}

{blackberry, blueberry, lemon}

A pizza place offers mushrooms, tomatoes and sausage as toppings for a plain cheese base. How many different types of pizzas can be made?

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

The set of possible toppings is { M , T , S }

A pizza place offers mushrooms, tomatoes and sausage as toppings for a plain cheese base. How many different types of pizzas can be made?

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

The set of possible toppings is { M , T , S }Remember: A set with k elements has subsets.

A pizza place offers mushrooms, tomatoes and sausage as toppings for a plain cheese base. How many different types of pizzas can be made?

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

The set of possible toppings is { M , T , S }

Every subset of this set is a different pizza.

Remember: A set with k elements has subsets.

A pizza place offers mushrooms, tomatoes and sausage as toppings for a plain cheese base. How many different types of pizzas can be made?

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

The set of possible toppings is { M , T , S }

Every subset of this set is a different pizza.So, there are

different types of pizzas possible.

Remember: A set with k elements has subsets.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Amber wants to visit Dallas, Reno, Tulsa, Orlando, Atlanta, Nashville, Phoenix, Mobile and Indianapolis. If she can decide to visit all, some or none of these cities, how many travel options does Amber have?

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Amber wants to visit Dallas, Reno, Tulsa, Orlando, Atlanta, Nashville, Phoenix, Mobile and Indianapolis. If she can decide to visit all, some or none of these cities, how many travel options does Amber have?

The set of possible cities is {D, R, T, O, A, N, P, M, I}

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Amber wants to visit Dallas, Reno, Tulsa, Orlando, Atlanta, Nashville, Phoenix, Mobile and Indianapolis. If she can decide to visit all, some or none of these cities, how many travel options does Amber have?

The set of possible cities is {D, R, T, O, A, N, P, M, I}321 4 5 6 7 8 9

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Amber wants to visit Dallas, Reno, Tulsa, Orlando, Atlanta, Nashville, Phoenix, Mobile and Indianapolis. If she can decide to visit all, some or none of these cities, how many travel options does Amber have?

The set of possible cities is {D, R, T, O, A, N, P, M, I}321 4 5 6 7 8 9

Remember: A set with k elements has subsets.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Amber wants to visit Dallas, Reno, Tulsa, Orlando, Atlanta, Nashville, Phoenix, Mobile and Indianapolis. If she can decide to visit all, some or none of these cities, how many travel options does Amber have?

The set of possible cities is {D, R, T, O, A, N, P, M, I}321 4 5 6 7 8 9

Every subset of this set is a different travel option.Remember: A set with k elements has subsets.

MATH 110 Sec 2-2: Comparing SetsPractice Exercises

Amber wants to visit Dallas, Reno, Tulsa, Orlando, Atlanta, Nashville, Phoenix, Mobile and Indianapolis. If she can decide to visit all, some or none of these cities, how many travel options does Amber have?

The set of possible cities is {D, R, T, O, A, N, P, M, I}321 4 5 6 7 8 9

So, Amber has512

different travel options.

Remember: A set with k elements has subsets.

Every subset of this set is a different travel option.