Math: Ch. 2

{..., -10, -9, -6, -3, -2, 2, 3, 6, 9, 10, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe (A ∪ B) − C by listing their elements explicitly.

{..., -12, -8, -4, 0, 4, 8, 12, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe (A ∪ B) ∩ C by listing their elements explicitly.

{..., -10, -8, -4, -2, 2, 4, 6, 10, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe A − B by listing their elements explicitly.

{..., -12, -6, 0, 6, 12, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe A ∩ B by listing their elements explicitly.

{..., -10, -8, -4, -2, 2, 4, 6, 10, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe A ∩ ~B by listing their elements explicitly.

{..., -9, -3, 3, 9, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe B − A by listing their elements explicitly.

{..., -12, -9, -8, -6, -4, -3, 0, 3, 4, 6, 8, 9, 12, ...}

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe B ∪ C by listing their elements explicitly.

∅

Assume U = Z, and let A = {. . ., −6, −4, −2, 0, 2, 4, 6, . . .} = 2Z, B = {. . ., −9, −6, −3, 0, 3, 6, 9, . . .} = 3Z, C = {. . ., −12, −8, −4, 0, 4, 8, 12, . . .} = 4Z. Describe C − A by listing their elements explicitly.

65536

Determine |℘(℘(℘({1, 2})))|

{2, 4}

Find a set of largest possible size that is a subset of both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.

{1, 2, 3, 4, 5, 6, 8, 10}

Find a set of smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets.

2

Find the least element of {n ∈ N | n = k² + 1 for some k ∈ N}, if there is one.

3

Find the least element of {n ∈ N | n² − 3 ≥ 2}, if there is one.

3

Find the least element of {n ∈ N | n² − 5 ∈ N}, if there is one.

1

Find the least element of {n² + 1 | n ∈ N}, if there is one.

{∅, {4}, {7}, {4, 7}}

Find the power set of {4, 7}

{∅, {a}, {b}, {a, b}}

Find the power set of {a, b}

{∅, {a}, {{b}}, {a, {b}}}

Find the power set of {a, {b}}

{∅, {x}, {y}, {z}, {w}, {x, y}, {x, z}, {x, w}, {y, z}, {y, w}, {z, w}, {x, y, z}, {x, y, w}, {x, z, w}, {y, z, w}, {x, y, z, w}}

Find the power set of {x, y, z, w}

{∅, {{a}}}

Find the power set of {{a}}

{∅, {{a}}, {{b}}, {{a}, {b}}}

Find the power set of {{x}, {y}}

8

Find |A ∩ B| when A = {x ∈ N | x ≤ 20} and B = {x ∈ N | x is prime}.

34

Find |A| when A = {4, 5, 6, . . ., 37}.

103

Find |A| when A = {x ∈ Z | −2 ≤ x ≤ 100}.

4

Let A = {1, 2, 3, 4, 5} and B = {2, 3, 4}. How many sets C have the property that C ⊆ A and B ⊆ C?

{1, 2}

Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 5}. Find A \ B.

{1}

Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 5}. Find A ∩ (B ∪ C).

{3, 4, 5}

Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 5}. Find A ∩ B.

{1, 2, 3, 4, 5, 6, 7}

Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 5}. Find A ∪ B.

11

Let A = {n ∈ N | 20 ≤ n < 50} and B = {n ∈ N | 10 < n ≤ 30}. Suppose C is a set such that C ⊆ A and C ⊆ B. What is the largest possible cardinality of C?

{5, 7, 9, 11}

Let A = {x ∈ N | 4 ≤ x < 12} and B = {x ∈ N | x is even}. Find A \ B.

{4, 6, 8, 10}

Let A = {x ∈ N | 4 ≤ x < 12} and B = {x ∈ N | x is even}. Find A ∩ B.

A = B

Let A and B be arbitrary nonempty sets. Under what condition does A × B = B × A?

A and B have completely different elements

Let A and B be arbitrary nonempty sets. Under what condition is (A × B) ∩ (B × A) empty?

{4, 9}

Let A ∈ {1, 4, 9} and B ∈ {1, 3, 6, 10}. Find A \ B.

{1}

Let A ∈ {1, 4, 9} and B ∈ {1, 3, 6, 10}. Find A ∩ B.

{1, 3, 4, 6, 9, 10}

Let A ∈ {1, 4, 9} and B ∈ {1, 3, 6, 10}. Find A ∪ B.

{3, 6, 10}

Let A ∈ {1, 4, 9} and B ∈ {1, 3, 6, 10}. Find B \ A.

{(-2, 0), (-2, 4), (2, 0), (2, 4)}

Let X = {−2, 2}, Y = {0, 4} and Z = {−3, 0, 3}. Evaluate X × Y

{(-2, -3), (-2, 0), (-2, 3), (2, -3), (2, 0), (2, 3)}

Let X = {−2, 2}, Y = {0, 4} and Z = {−3, 0, 3}. Evaluate X × Z

{(-3, 0, 0), (-3, 0, 4), (-3, 4, 0), (-3, 4, 4), (0, 0, 0), (0, 0, 4), (0, 4, 0), (0, 4, 4), (3, 0, 0), (3, 0, 4), (3, 4, 0), (3, 4, 4)}

Let X = {−2, 2}, Y = {0, 4} and Z = {−3, 0, 3}. Evaluate Z × Y × Y

~B

Let the universal set U be the set of people who voted in the 2012 U.S. presidential election. Define the subsets D, B, and W of U as follows: D = {x ∈ U | x registered as a Democrat}, B = {x ∈ U | x voted for Barack Obama}, W = {x ∈ U | x belonged to a union}. Express the following subset of U in terms of D, B, and W: People who did not vote for Barack Obama.

B ∩ ~D ∩ ~W

Let the universal set U be the set of people who voted in the 2012 U.S. presidential election. Define the subsets D, B, and W of U as follows: D = {x ∈ U | x registered as a Democrat}, B = {x ∈ U | x voted for Barack Obama}, W = {x ∈ U | x belonged to a union}. Express the following subset of U in terms of D, B, and W: People who voted for Barack Obama but were not registered as Democrats and were not union members.

(D ∩ W) U ~B

Let the universal set U be the set of people who voted in the 2012 U.S. presidential election. Define the subsets D, B, and W of U as follows: D = {x ∈ U | x registered as a Democrat}, B = {x ∈ U | x voted for Barack Obama}, W = {x ∈ U | x belonged to a union}. Express the following subset of U in terms of D, B, and W: People who were either registered as Democrats and were union members or did not vote for Barack Obama.

(D ∩ B) - ~W

Let the universal set U be the set of people who voted in the 2012 U.S. presidential election. Define the subsets D, B, and W of U as follows: D = {x ∈ U | x registered as a Democrat}, B = {x ∈ U | x voted for Barack Obama}, W = {x ∈ U | x belonged to a union}. Express the following subset of U in terms of D, B, and W: Registered Democrats who voted for Barack Obama but did not belong to a union.

W ∩ (~D U B)

Let the universal set U be the set of people who voted in the 2012 U.S. presidential election. Define the subsets D, B, and W of U as follows: D = {x ∈ U | x registered as a Democrat}, B = {x ∈ U | x voted for Barack Obama}, W = {x ∈ U | x belonged to a union}. Express the following subset of U in terms of D, B, and W: Union members who either were not registered as Democrats or voted for Barack Obama.

W ∩ B

Let the universal set U be the set of people who voted in the 2012 U.S. presidential election. Define the subsets D, B, and W of U as follows: D = {x ∈ U | x registered as a Democrat}, B = {x ∈ U | x voted for Barack Obama}, W = {x ∈ U | x belonged to a union}. Express the following subset of U in terms of D, B, and W: Union members who voted for Barack Obama.

B ∩ C

Prove that if A ⊆ B and A ⊆ C, then A ⊆ B ∩ C. x ∈ ???

False

True or false: 1} ⊆ {{1}, {1, 2}}

False

True or false: If A = {1, 2, 3}, then {1} is a subset of ℘(A).

True

True or false: The empty set ∅ is a subset of {1, 2, 3}.

False

True or false: [1, 2) ∪ (2, 3] = [2, 3]

False

True or false: [1, 2] ∩ [2, 3] = ∅

True

True or false: [2, 4) ⊂ [2, 4]

False

True or false: [2, 4) ⊆ (2, 4]

True

True or false: [2, 4] ⊆ (0, 6)

False

True or false: [3, 4) ⊆ (3, 4)

True

True or false: a ∈ {a}

False

True or false: a ⊆ {a}

True

True or false: {1, 2, 3} ⊆ N

True

True or false: {1, 2, 3} ⊆ {0, 1, 2, 3, 4}

True

True or false: {1, 2} ⊂ [1, 2]

True

True or false: {1} ⊆ {1, {1, 2}}

True

True or false: {3, 5} = {5, 3}

True

True or false: {a} ⊆ {a, b}

False

True or false: {a} ⊆ ℘({{a}, {b}})

True

True or false: {} = ∅

False

True or false: ∅ = {∅}

True

True or false: ∅ ∈ {∅}

False

True or false: ∅ ∈ ∅

True

True or false: ∅ ⊂ {∅}

True

True or false: ∅ ⊆ {∅}

True

True or false: ∅ ⊆ ∅

{3, 6, 9, 12, 18}

Use the roster method to describe {x ∈ N | x < 20 and x is a multiple of 3 but not a multiple of 5}

{3, 6, 9, 12, 15, 18}

Use the roster method to describe {x ∈ N | x < 20 and x is a multiple of 3}

{-15, 0, 15}

Use the roster method to describe {x ∈ Z | |x| < 20 and x is a multiple of 3 and a multiple of 5}

{-18, -15, -12, -10, -9, -6, -5, -3, 0, 3, 5, 6, 9, 10, 12, 15, 18}

Use the roster method to describe {x ∈ Z | |x| < 20 and x is a multiple of 3 or a multiple of 5}

∅

Write (4, 5) ∩ Z by listing its elements explicitly.

{-3, -2, -1, 0, 1, 2, 3, 4}

Write (−4, 4] ∩ Z by listing its elements explicitly.

{-3, -2, -1, 0, 1, 2, 3, 4, ...}

Write (−4, ∞) ∩ Z by listing its elements explicitly.

{-3, -2, -1}

Write (−4, ∞) ∩ Z⁻ by listing its elements explicitly.

{1, 2, 3, 4}

Write (−∞, 4] ∩ N by listing its elements explicitly.

{-4, -3, -2, -1, 0, 1, 2, 3, 4}

Write [−4, 4] ∩ Z by listing its elements explicitly.

{n ∈ Z | n = 6k - 2 for some integer k}

Write {. . ., −14, −8, −2, 4, 10, 16, . . .} in the form {n ∈ Z | p(n)} with a logical statement p(n) describing the property of n.

{n ∈ Z | n = 5k for some integer k}

Write {. . ., −15, −10, −5, 0, 5, 10, 15, . . .} in the form {n ∈ Z | p(n)} with a logical statement p(n) describing the property of n.

{n ∈ Z | n = k³ for some integer k}

Write {. . ., −27, −8, −1, 0, 1, 8, 27, . . .} in the form {n ∈ Z | p(n)} with a logical statement p(n) describing the property of n.

{n ∈ Z | n < 0}

Write {. . ., −3, −2, −1} in the form {n ∈ Z | p(n)} with a logical statement p(n) describing the property of n.

{n ∈ Z | n = k² for some integer k}

Write {0, 1, 4, 9, 16, . . .} in the form {n ∈ Z | p(n)} with a logical statement p(n) describing the property of n.

{n ∈ Z | n ≥ 0 and n = 4k for some integer k}

Write {0, 4, 8, 12, . . .} in the form {n ∈ Z | p(n)} with a logical statement p(n) describing the property of n.

{1, 2, 3}

Write {n ∈ N | −6 < n < 4} by listing its elements explicitly (that is, using the roster method).

{-5, -4, -3, -2, -1, 0, 1, 2, 3}

Write {n ∈ Z | −6 < n < 4} by listing its elements explicitly (that is, using the roster method).

{-2, 0, 3}

Write {x ∈ Q | x³ − x² − 6x = 0} by listing its elements explicitly (that is, using the roster method).

{-3, 3}

Write {x ∈ Q | x⁴ − 11x² + 18 = 0} by listing its elements explicitly (that is, using the roster method).