Discrete Math: Functions and Relations - D421 exam | Full Questions, Correct Answers, and Worked Solutions | 2026 Update | 100% Correct.

Discrete Math:
Functions and Relations
- D421 exam | Full
Questions, Correct
Answers, and Worked
Solutions | 2026 Update
| 100% Correct.

C




Terms in this set (260)



N The set of natural numbers: All integers greater than
or equal to 0.


e.g. 0, 1, 2, 3, ...


Z The set of all integers (positive, negative, and zero).


e.g. -2, -1, 0, 1, 2, ...


Q The set of rational numbers: All real numbers that
can be expressed as a/b, where a and b are integers
and b ≠ 0.


e.g. 0, 1/2, 5.23, -5/3

R The set of real numbers: Includes all rational and
irrational numbers.


e.g. 0, 1/2, 5.23, -5/3, π, √2

,cardinality The cardinality of a finite set A, denoted by |A|, is the
number of elements in A. If A = { 2, 4, 6, 10 }, then |A|
= 4. The cardinality of the set B = { 1, 3, 5, ... , 99 } is
50. The cardinality of the empty set |∅| is zero.


∅ The set with no elements is called the empty set and is
denoted by the symbol ∅. The empty set is
sometimes referred to as the null set and can also be
denoted by {}. Because the empty set has no
elements, for any element a, a ∉
∅ is true.

elements The objects in a set are called elements.


set A set is a collection of objects


universal set The universal set, written as U, is the big set that
includes
everything you're talking about in a specific situation.


🧠 Think of it like this:
It’s the “universe” of your discussion.
Every other set you mention is a smaller part of this big
set.


🧪 Examples:
If you're talking about types of real numbers, then U
= all real numbers.
If you're talking about students' grades at a school,
then U = all students at that school.

Subset (⊆) Definition:
Set A is a subset of B if every element in A is
also in B. A can be equal to B.


Example:
A = {1, 2, 3}, B = {1, 2, 3} → A ⊆ B


Proper Subset (⊂) Definition:
Set A is a proper subset of B if every element in A is
in B, and A is not equal to B.


Example:
A = {1, 2}, B = {1, 2, 3} → A ⊂ B


Power Set (P(A)) The power set of a set A, written as P(A), is the set of
all subsets of A. This includes:
The empty set (∅)
All individual elements as
sets All combinations of
elements The set A itself
Key Facts:
∅ is always in P(A)
If A has n elements, then P(A) has 2ⁿ subsets
Example: If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}} |P(A)| =
4 = 2²

, Intersection of Sets (A ∩ B) The intersection of sets A and B is the set of all
elements that are in both A and B. It's written as A ∩ B
and read as "A intersect B."


Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}


Key Idea: Only the elements that A and B share go
into the intersection.



Union of Sets (A ∪ B) The union of two sets A and B, written as A ∪ B and
read "A union B," is the set of all elements that are in
A, in B, or in both. It uses the inclusive OR, meaning
that if an element is in both sets, it still appears in
the union.


Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2,
3, 4,
5}


Key Idea: Union combines everything from both sets
without duplicates.

Set Difference (A − B) The difference between two sets A and B, written as A
− B, is the set of elements that are in A but not in B.
This operation removes any elements that A and B
share.


Example: If A = {1, 2, 3, 4} and B = {3, 4, 5}, then A − B = {1,
2}


Key Idea:
A − B ≠ B − A (Set difference is not
commutative) Think of it as "subtracting" B
from A

Symmetric Difference (A ⊕ B) The symmetric difference between two sets A and B,
written as A ⊕ B, is the set of elements that are in
exactly one of the sets — not both. It is commutative,
meaning A ⊕ B = B ⊕ A.


Alternative Definition: A ⊕ B = (A − B) ∪ (B − A) This
means: take the elements only in A, and the elements
only in B, then combine them.


Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ⊕ B = {1,


2, 4, 5} Key Idea:
Removes shared elements
Keeps only the non-overlapping parts
Great for comparing differences without duplication