WGU C959 Modified latest updated version with 100% correct answers 2026-2027

What is the notation for subset?

⊆

A collection of objects is known as a

set

What is the notation for integers?

ℤ

What is the notation for element?

∈

What is the notation for set-roster?

{ }

Write the set-builder notation for, "Numbers whose square roots are an integer"

{ x | √x ∈ ℤ}

Write the set-builder notation for "the set of all x's, such that x is greater than 0"

{ x | x > 0}

T/F:

Order matters in ordered pairs?

True

In a Cartesian Product of two sets, every element of the CROSS PRODUCT is an

ordered pair

What is the Cartesian Product for:

{a,b} x {0,1}

A x B = { {a,1} , {a,0} , {b,1} , {b,0} }

Ordered pairs are _____ of the Cartesian Product

elements

A ______ is a subset between two different sets

relation

Give the general equation for Relations

(a,b) ∈ A x B

(i.e. (a,b) is the ordered pair and A x B are two different sets)

Describe in words what this formula means: (a,b) ∈ A x B

Ordered pair (a,b) are in the set resulting from A x B

A compound proposition is a tautology if the proposition is always _____

True

A compound proposition is a contradiction if the proposition is always _____

False

If p is False and q is True, solve this equation.

p → q

True

If an equation is show as this "p → q" and the hypothesis is false, then the answer to the question is _____

True

Give the truth table for ¬(p ↔ q)

F

T

T

F

What is the logical equivalence of ¬(p ∧ q) ≡ ?

(¬p ∨ ¬q)

What is the logical equivalence of ¬(p ∨ q) ≡ ?

(¬p ∧ ¬q)

What is the logical equivalence of p→q ≡ ?

(¬p ∨ q)

In Boolean Algebra the addition symbol is the same as what?

OR

The XOR operation outputs 1 when what?

Both inputs are different

(1 XOR 0 = 1)

(1 XOR 1 = 0)

Boolean multiplication is the same as what?

AND

The minterm must evaluate to what

1

0 NAND 1 =

1

1 NAND 1 =

0

0 NAND 0 =

1

The NAND gate computes the NAND operation:

x↑y

The NOR gate computes the NOR operation:

x↓y

The NAND gate outputs 0 if all inputs are _____

1

The NOR gate outputs 1 if all inputs are _____

0

The gate outputs 1 if all inputs are 0 and outputs _____

0

1 NOR 1 =

0

1 NOR 0 =

0

0 NOR 0 =

1

A two-input XOR gate (for "exclusive OR") outputs 1 if the input values differ. True or False

True

1 XOR 0 =

1

1 XOR 1 =

0

0 XOR 0 =

0

A two-input XNOR gate (for "exclusive NOR") outputs 1 if the input values are the same. True or False

True

1 XNOR 0 =

0

1 XNOR 1 =

1

0 XNOR 0 =

1

Which gate follows the same rules as Boolean multiplication?

AND

Which gate follows the same rules as Boolean addition?

OR

What is the maximum length of a cycle in a graph?

The amount of Vertices that connect.

The proposition p ⊕ q is true if...

Exactly one of the propositions p and q is true but not both

Give DeMorgans Law for QUANTIFIED STATEMENTS (there are 2 laws)

¬∀xP(x) ≡ ∃x¬P(x)

¬∃xP(x) ≡ ∀x¬P(x)

Use DeMorgans Law to solve: ¬∃x P(x)

∀x ¬P(x)

Use DeMorgans Law to solve: ¬∃x (P(x) ∨ Q(x))

∀x (¬P(x) ∧ ¬Q(x))

Use DeMorgans Law to solve: ¬∀x (P(x) ∧ Q(x))

∃x (¬P(x) ∨ ¬Q(x))

Is the variable y bound in the expression ∀xQ(x,y)?

No

Is the following logical expression a proposition: ∀z∃yQ(x,y,z)?

Why?

No. X is not bound

If the domain of a universal statement is small, it may be easiest to prove the statement by checking each element individually. A proof of this kind is called a _____

proof by exhaustion

A _____ is an assignment of values to variables that shows that a universal statement is false

counterexample

When are XNOR gates equal to 0?

When both inputs = 0

When both inputs = 1

The NAND operation outputs 1 for all combinations except for _____

1 NAND 1

contrapositive of a conditional statement

If not q, then not p

In a _______ conditional statement we assume the question is false, and then use the falsity to prove that the statement is possibly true

contradiction

Because the empty set has no elements, for any element a, a ∉ ∅ is _____

true

The number of elements in a set is referred to as the _____

cardinality

What is the cardinality of the set A = { 2, 4, 6, 10 }

|A| = 4

What is the cardinality of B = { 1, 3, 5, ... , 99 }

|B| = 50

Two sets are equal if they have exactly the same ______

elements

What is the notation for "The set of natural numbers:" (All integers greater than or equal to 0.)

N

What is the notation for "The set of rational numbers:"

Q

What is a rational number?

A number that can be written as a fraction, where the denominator does NOT equal 0

What is the notation for "The set of real numbers."

R

T/F:

-3 ∈ Z+

False

T/F:

0 ∈ Z+

False

T/F:

0 is a non-negative integer

True

T/F:

5 ∈ R+

True

T/F:

0 ∈ Q

True (0 is a rational number)

The only real numbers that satisfy |x| = x2 are what?

-1, 0, 1

What set matches this set: { x ∈ R : |x| = x2 }

{-1, 0 1}

If every element in A is also an element of B, then A is a ______ of B

subset

If there is an element of A that is not an element of B, then A is not a subset of B, denoted as A ⊈ B. If the universal set is U, then for every set A: (Write the relationship for this)

∅ ⊆ A ⊆ U

If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a ______ of B

proper subset

What is the notation for proper subset?

⊂

If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a proper subset of B, denoted as _____

A ⊂ B

A = { 3, 4, 5 }

B = { 4, 5, 3 }

A ⊆ B?

True or False?

True

A = { 3, 4, 5 }

B = { 4, 5, 3 }

A ⊂ B?

True or False?

False (the sets are equal, so A is not a proper subset of B)

C = { x ∈ Z: x is odd }

C ⊂ Z?

True or False

True (odd integers are a subset of all integers)

B = { 4, 5, 3 }

D = { 3, 5, 7, 9 }

B ⊆ D?

True or False

False (4 is an element of B, but 4 is not an element of D)

Is the following statement true?For any two sets, X and Y, if X ⊂ Y, then X ⊆ Y.

Yes

No

Yes (If X ⊂ Y, then every element of X is also an element of Y, so X ⊆ Y.)

Is the following statement true?For any two sets, X and Y, if X ⊆ Y, then X ⊂ Y.

Yes

No

No (It is possible that X = Y in which case X ⊆ Y, but X ⊄ Y)

A = { x ∈ Z: x is an integer multiple of 3 }

E = { 3, 6, 9 }

E ⊆ A?

True or False

True

A = { x ∈ Z: x is an integer multiple of 3 }

E = { 3, 6, 9 }

A ⊂ E?

True or False

False

A = { x ∈ Z: x is an integer multiple of 3 }

E = { 3, 6, 9 }

E ∈ A?

True or False

False (E is not an element of A. All the elements of A are numbers, not sets. However, it is true that E ⊂ A)

A = { x ∈ Z: x is an integer multiple of 3 }

15 ⊂ A?

True or False

False (15 is not a set, so 15 can not be a subset of any set)

A = { x ∈ Z: x is an integer multiple of 3 }

{15} ⊂ A?

True or False

True (15 is now considered a set because of the brackets around 15, so this statement is true)

A = { x ∈ Z: x is an integer multiple of 3 }

∅ ⊂ A?

True or False

True (The empty set is a subset of every set)

A = { x ∈ Z: x is an integer multiple of 3 }

A ⊆ A?

True or False

True (Every set is a subset of itself)

B = { x ∈ Z: x is a perfect square }

∅ ∈ B?

True or False

False (All elements in B are numbers, not sets)

E = { 3, 6, 9 }

F = { 4, 6, 16 }

|E| = |F|?

True or False

True (the cardinality of F and E are the same)

B = { x ∈ Z: x is a perfect square }

B is a finite set?

True or False

False (There is an infinite number of perfect squares)

A = {2, 4, 6, 8}

B = {x ∈ Z: x is even and 0 < x < 10}

A ⊂ B?

True or False

False (A = B. There is no element of B that is not also an element of A)

A = {2, 4, 6, 8}

C = {x ∈ Z: x is even and 0 < x ≤ 10}

A ⊂ C?

True or False

True (Every element of A is also an element of C. Also 10 ∈ C and 10 ∉ A)