WGU C959 - Discrete Math 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 two sets A x B

True ∧ True =

True

True ∧ False =

False

False ∧ False =

False

True ∨ False =

True

True ∨ True =

True

False ∨ False =

False

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

Contradiction, Contrapositive or Direct Proof:

p→q = p→-q

Contrapositive

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

true

The set with no elements is called the empty set and is denoted by the symbol:

∅

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)