Discrete Mathematics - Chapter 6: Set Theory

Given a universal set U, set A and set B (using Venn Diagram): P (A ∪ B) = P(A) .... P(B) ..... P(A ∩ B)

+, -

Which one the symbolically of "The union of A and B"? 1. A ∪ B = {x ∈ U | x ∈ A or x ∈ B} 2. A ∩ B = {x ∈ U | x ∈ A and x ∈ B} 3. B − A = { x ∈ U | x ∈ B a n d x ⊄ A } 4. A'= { x ∈ U | x ⊄ A } .

1

{x ∈ U | x ∈ A or x ∈ B} = ..... 1. A ∪ B 2. A ∩ B 3. B - A 4. A'

1

{ x ∈ U | x ∈ B a n d x ⊄ A } = ..... 1. A ∪ B 2. A ∩ B 3. B - A 4. A'

3

Given a universal set U, set A and set B (using Venn Diagram): A ∪ A' = ......

U

Given a universal set U, set A and set B (using Venn Diagram): A' ∩ B' = ......

U

Let the universal set be the set U = {a,b,c,d,e, f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. => B - A = { ............ }

d, f

Let A = {1,2,3,4,5,6}, A1 = {1,2}, A2 = {3,4}, and A3 = {5,6}. True of False? "{A1, A2, A3} a partition of A"

true

Element Argument: The Basic Method for Proving That One Set Is a Subset of Another: Let sets X and Y be given. To prove that X ⊆Y: 1. suppose that ..... is a particular but arbitrarily chosen element of X, 2. show that x is an element of ........

x, Y

Let A = {1} and B = {1, {1}}. 1. Is A ⊆ B? 2. If so, is A a proper subset of B?

yes, yes

A₁ × A₂ = ...... 1. { (a₁, a₂) | a₁ ∈ A₁ and a₂ ∈ A₂} 2. { (a₁, a₂) | a₁ ∈ A₁ and a₂ ∉ A₂} 3. { (a₁, a₂) | a₁ ∉ A₁ and a₂ ∈ A₂} 4. { (a₁, a₂) | a₁ ∈ A₂ and a₂ ⊄ A₁}

1

Find the power set of the set {x, y}. That is, find P({x, y}): 1. P({x, y}) = {∅, {x}, {y}, {x, y}}. 2. P({x, y}) = {∅, {x}, {x, y}}. 3. P({x, y}) = {{x}, {y}, {x, y}}. 4. P({x, y}) = {∅, {x}, {y}}.

1

Given real numbers a and b with a ≤ b: {x ∈ R | a < x < b} = ...... 1. (a, b) 2. (a, b] 3. [a, b] 4. [a, b)

1

Given real numbers a and b with a ≤ b: {x ∈ R| x > a} = ...... 1. (a,∞) 2. (−∞,b) 3. [a,∞) 4. [−∞,b)

1

Given sets A₀, A₁, A₂, . . . that are subsets of a universal set U and given a nonneg- ative integer n: {x ∈ U|x ∈ Aι for at least one ι = 0,1,2,...,n} =...... 1. the union of the A-sub-ι from ι equals 0 to n 2. the union of the A-sub-i from i equals 0 to ∞ 3. the intersection of the A-sub-i from i equals 0 to n 4. the intersection of the A-sub-i from i equals 0 to ∞

1

The formula to find the power set P(A) of a set A with n elements: 1. 2ⁿ 2. 2ⁿ⁻¹ 3. 2n 4. 2n - 1

1

Which one is correct? 1. A ⊄ B ⇔ ∃x such that x ∈ A and x ⊄ B. 2. A ⊄ B ⇔ ∃x such that x ∈ B and x ⊄ A. 3. B ⊄ A ⇔ ∃x such that x ∈ A and x ⊄ B. 4. B ⊄ A ⇔ ∃x such that x ∈ B and x ⊄ A.

1, 4

(a, b) = (c, d) ⇔ .... = c and ..... = d.

a, b

Let the universal set be the set U = {a,b,c,d,e, f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. => B' = { ............ }

a, b, c

Let the universal set be the set U = {a,b,c,d,e, f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. => A ∪ B = { ............ }

a, c, d, e, f, g

1. x ∉ A ∪ B if, and only if, x ∉ A ...... x ∉ B 2. x ∉ A ∩ B if, and only if, x ∉ A ...... x ∉ B 3. x ∉ A - B if, and only if, x ∉ A ∩ ...... 4. x ∉ A - B if, and only if, x ∉ A ..... x ∈ B

and, or, B', or

Let the universal set be the set U = {a,b,c,d,e, f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. => A' = { ............ }

b, d, f

The ....... of A, denoted A', is the set of all elements in U that are not in A.

complement

The ........ of B minus A (or relative complement of A in B), denoted B− A, is the set of all elements that are in B and not A.

difference

Two sets are called ....... if, and only if, they have no elements in common.

disjoint

Let the universal set be the set U = {a,b,c,d,e, f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. => A ∩ B = { ............ }

e, g

The ...... set (or null set) is denoted by the symbol ∅

empty

Given a universal set U, set A and set B (using Venn Diagram): True or False? "A ∩ A' = U"

false

Let A = {1,2,3,4,5,6}, A1 = {1,2}, A2 = {3,4,5}, and A3 = {5,6}. True of False? "{A1, A2, A3} a partition of A"

false

The ....... of A and B, denoted A ∩ B, is the set of all elements that are common to both A and B.

intersection

Sets A₁, A₂, A₃ . . . are ........... if, and only if, no two sets Aι and Aκ with distinct subscripts have any elements in common. More precisely, for all ι, κ = 1, 2, 3, . . .

mutually disjoint

A finite or infinite collection of nonempty sets {A₁, A₂, A₃ ...} is a ...... of a set A if, and only if: 1. A is the union of all the Aκ 2. The sets A₁, A₂, A₃, . . . are mutually disjoint.

partition

Given a set A, the ....... set of A, denoted P(A), is the set of all subsets of A.

power

A is a ........ of B ⇔ (1) A ⊆ B (2) there is at least one element in B that is not in A.

proper subset

The symbols ∞ and −∞ are used to indicate intervals that are unbounded either on the ..... or on the ......:

right, left

a set A to be a ....... of a set B as a formal universal conditional statement: A ⊆ B ⇔ ∀x,if x ∈ A then x ∈ B.

subset

Given a universal set U, set A and set B (using Venn Diagram): True or False? "A ∩ A' = ∅"

true

True or False? if : 1. A = {m ∈ Z | m = 2a for some integer a} 2. B = {n ∈ Z | n = 2b − 2 for some integer b} Then: A = B

true

True or False? "A and B are disjoint ⇔ A ∩ B = ∅."

true

True or False: a. Z⁺⊆ Q b. R⁻ ⊆ Q c. Q ⊆ Z d. Z⁻ ∪ Z⁺ = Z e. Z⁻ ∩ Z⁺ = ∅ f. Q ∩ R = Q g. Q ∪ Z = Q h. Z⁺ ∩ R = Z⁺ i. Z ∪ Q = Z

true, false, false, false, true, true, true, true, false

The ...... of A and B, denoted A ∪ B, is the set of all elements that are in at least one of A or B.

union

Given real numbers a and b with a ≤ b: {x ∈ R| x ≤ b} = ...... 1. (a,∞) 2. (−∞,b) 3. [a,∞) 4. [−∞,b)

4

Given sets A₀, A₁, A₂, . . . that are subsets of a universal set U and given a nonneg- ative integer n: {x ∈ U | x ∈ Aι for all nonnegative integers ι = 0,1,2,...,n } = ............ 1. the union of the A-sub-ι from ι equals 0 to n 2. the union of the A-sub-i from i equals 0 to ∞ 3. the intersection of the A-sub-i from i equals 0 to n 4. the intersection of the A-sub-i from i equals 0 to ∞

4

Which one the symbolically of "The complement of A"? 1. A ∪ B = {x ∈ U | x ∈ A or x ∈ B} 2. A ∩ B = {x ∈ U | x ∈ A and x ∈ B} 3. B − A = { x ∈ U | x ∈ B a n d x ⊄ A } 4. A'= { x ∈ U | x ⊄ A } .

4

{ x ∈ U | x ∈/ A } = ..... 1. A ∪ B 2. A ∩ B 3. B - A 4. A'

4

Given a universal set U, set A, set B and set C: How many areas (in Venn Diagram) can be created?

6

Given a universal set U, set A and set B (using Venn Diagram): A ∩ B' = ......

A

..... = ..... ⇔ A ⊆ B and B ⊆ A.

A, B

....... ⊆ ...... ⇔ ∀x,if x ∈ A then x ∈ B.

A, B

A ⊆ B ⇔ ∀x,if x ∈ ..... then x ∈ ........

A, B

Given a universal set U, set A and set B (using Venn Diagram): B ∩ A' = ......

B

Given a universal set U, set A and set B: How many areas (in Venn Diagram) can be created?

4

Given real numbers a and b with a ≤ b: {x ∈ R | a ≤ x < b} = ...... 1. (a, b) 2. (a, b] 3. [a, b] 4. [a, b)

4

Given real numbers a and b with a ≤ b: {x ∈ R | a < x ≤ b} = ...... 1. (a, b) 2. (a, b] 3. [a, b] 4. [a, b)

2

Given real numbers a and b with a ≤ b: {x ∈ R| x < b} = ...... 1. (a,∞) 2. (−∞,b) 3. [a,∞) 4. [−∞,b)

2

Given sets A₀, A₁, A₂, . . . that are subsets of a universal set U and given a nonneg- ative integer n: {x ∈ U | x ∈ Aι, for at least one nonnegative integer ι } = ............ 1. the union of the A-sub-ι from ι equals 0 to n 2. the union of the A-sub-i from i equals 0 to ∞ 3. the intersection of the A-sub-i from i equals 0 to n 4. the intersection of the A-sub-i from i equals 0 to ∞

2

Which one is equal to A = B? 1. A ⊄ B and B ⊆ A. 2. A ⊆ B and B ⊆ A. 3. A ⊆ B and B ⊄ A.

2

Which one the symbolically of "The intersection of A and B"? 1. A ∪ B = {x ∈ U | x ∈ A or x ∈ B} 2. A ∩ B = {x ∈ U | x ∈ A and x ∈ B} 3. B − A = { x ∈ U | x ∈ B a n d x ⊄ A } 4. A'= { x ∈ U | x ⊄ A } .

2

{x ∈ U | x ∈ A and x ∈ B} = ..... 1. A ∪ B 2. A ∩ B 3. B - A 4. A'

2

Given real numbers a and b with a ≤ b: {x ∈ R | a ≤ x ≤ b} = ...... 1. (a, b) 2. (a, b] 3. [a, b] 4. [a, b)

3

Given real numbers a and b with a ≤ b: {x ∈ R| x ≥ a} = ...... 1. (a,∞) 2. (−∞,b) 3. [a,∞) 4. [−∞,b)

3

Given sets A₀, A₁, A₂, . . . that are subsets of a universal set U and given a nonneg- ative integer n: {x ∈ U | x ∈ Aι for all ι = 0,1,2,...,n} = ............ 1. the union of the A-sub-ι from ι equals 0 to n 2. the union of the A-sub-i from i equals 0 to ∞ 3. the intersection of the A-sub-i from i equals 0 to n 4. the intersection of the A-sub-i from i equals 0 to ∞

3

Which one the symbolically of "The difference of B and A"? 1. A ∪ B = {x ∈ U | x ∈ A or x ∈ B} 2. A ∩ B = {x ∈ U | x ∈ A and x ∈ B} 3. B − A = { x ∈ U | x ∈ B a n d x ⊄ A } 4. A'= { x ∈ U | x ⊄ A } .

3