The continuation of the first part.

1. SETS AND THE REAL
NUMBER SYSTEM
A Crash Course for Algebra Dummies

2. Lesson Two: Set Operations
How can something so simple be so f*cking
complicated? This is like addition and
subtraction, just different terms used.

3. Recap of Defined Terms
A SET is a well-defined collection of objects.

The members or objects are called

ELEMENTS.
F = {cunt, vagina, pussy}
A UNIVERSAL SET is the set of all elements

in a particular discussion.
The CARDINALITY of a set states the number

of elements a set contains. Denoted by n(A).
EQUAL SETS are sets with the exact same

elements.

4. Set Unions
Two sets can be quot;addedquot; together. The UNION

of two sets, denoted by ∪ , is the set of all
elements either in the first set, OR the other.
{1, 2} ∪ {red} = {1, 2, red}
{x, y, z} ∪ {1, 2} = {x, y, z, 1, 2}
The UNION of two sets is just a combination of

elements from both sets. Thus, A ∪ B is a
combination of the elements from both sets A
and B.

5. Basic Properties of Unions
A ∪ B = B ∪ A. Commutative Property of
1.
Unions.
A ∪ (B ∪ C) = (A ∪ B) ∪ C. Associative
2.
Property of Unions.
A ⊆ (A ∪ B). Obviously, since A ∪ B contains
3.
A.
A ∪ A = A. Uniting equal sets would result in
4.
the same set.
A ∪ ∅ = A. Identity Property of Unions.
5.
A ⊆ B if and only if A ∪ B = B. In this case, A
6.
= B, meaning A is an improper subset of B.

6. Set Intersections
The INTERSECTION of two sets, denoted by

∩ , is the set of elements that are members of
the first set AND the second set.
{1, 2} ∩ {red, white} = ∅.
{p, u, s, y} ∩ {f, u, c, k} = {u}.
The INTERSECTION of two sets is just getting

the common elements from both sets. Thus, A
∩ B is a set of elements found in A and also
found in B.

7. Basic Properties of
Intersections
A ∩ B = B ∩ A. Commutative Property of
1.
Intersections.
A ∩ (B ∩ C) = (A ∩ B) ∩ C. Associative
2.
Property of Intersections.
A ∩ B ⊆ A. True, since A has a part that B has.
3.
A ∩ A = A. Same elements from both sets.
4.
A ∩ ∅ = ∅. Zero Property of Intersections.
5.
A ⊆ B if and only if A ∩ B = A. In this case, A
6.
= B, following from the 3rd property of
intersections.

8. Exercise 1: Unions &
Intersections
Given the following sets:
F = {s, t, r, a, i, g, h} U = {g, a, y}
C = {l, e, s, b, i, a, n} K = Ø or { }
Find the following sets:
1. F ∪ U 2. C ∪ K 3. F ∪ U ∪ C
4. U ∩ K 5. F ∩ C 6. F ∩ U ∩ C
7. (F ∩ C) ∪ (U ∪ K) 8. (C ∪ U) ∩ (F ∩ K)

9. Answers 1: Unions &
Intersections
F ∪ U = {s, t, r, a, i, g, h, y}
1.
C ∪ K = {l, e, s, b, i, a, n}
2.
F ∪ U ∪ C = {s, t, r, a, i, g, h,
3.
F = {s, t, r, a, i, g, h}
y, l, e, b, n}
U = {g, a, y}
U∩K=Ø
4.
C = {l, e, s, b, i, a, n}
F ∩ C = {s, a, i}
5.
K = Ø or { }
F ∩ U ∩ C = {a}
6.
(F ∩ C) ∪ (U ∪ K) = {s, a, i, g,
7.
y}
(C ∪ U) ∩ (F ∩ K) = Ø
8.

10. Set Differences
Two sets can be “subtractedquot; too. The

DIFFERENCE of two sets, denoted by , is the
set of all elements left when the other
elements from the other set are removed.
{1, 2} {red, white} = {1, 2}.
{1, 2, green} {green} = {1, 2}.
{1, 2, 3, 4} {1, 3} = {2, 4}.

11. Set Differences
In DIFFERENCES, the trick is to remove all the

common elements of the first set and the
second set, then get the first set.
A = {1, 2, 3, 4} and B =
{1, 3, 5}
AB {1, 2, 3, 4}{1, 3, 5}
{2, 4}{5}
 AB = {2, 4}

12. Set Complements
The COMPLEMENT of a set is the set of all

elements that are not included in the set, but
are included in the universal set. It is denoted
by an apostrophe (‘). Given that:
U = {RE5, SH5, DMC4, GH:M}.
If A = {RE5, SH5},
Then A’ = {DMC4, GH:M}.

13. Basic Properties of
Complements
The trick in COMPLEMENTS is to get the

unmentioned elements in a set to form the new
one.
(A’)’ = A. Involution Property of Unions.
1.
U’= Ø. All elements are included in the
2.
universal set.
Under the same premise, Ø’ = U.
3.
A ∪ A’ = U. A’ is all members not part of A in
4.
the universal set.
Under the same premise, A ∩ A’ = Ø.
5.
AA = Ø. Obvious, since these are equal sets.
6.

14. Complements & Differences
Given the following sets:
U = {m, o, t, h, f, u, c, k, e, r} Q = {f, u, c,
k}
P = {r, o, t, c, h} E = {m, e, t, h} D = Ø or {
}
5. (P ∪ Q)’ ∪ E
1. Q P
Find the following 6. (PE) ∩ Q
sets:
2. (EP)Q
3. (PQ) ∩ E 7. (P ∪ E ∪ Q)’
8. (D ∩ U)’ ∪ Q
4. Q’P

15. Complements and Differences
Q P = {f, u, k}
U = {m, o, t, h, f, u, c, k, e, r}
1.
(EP)Q = {m, e}
2.
P = {r, o, t, c, h}
(PQ) ∩ E ={t, h}
E = {m, e, t, h}
Q = {f, u, c, k}
3.
D = Ø or { } Q’P ={m, e, r}
4.
(P ∪ Q)’ ∪ E = {m, e, t, h} or
5.
E
(PE) ∩ Q = {c}
6.
(P ∪ E ∪ Q)’ = Ø
7.
(D ∩ U)’ ∪ Q =
8.
{m, o, t, h, f, u, c, k, e, r} or

16. Cartesian Products
{1, 2} × {red, white} =
{(1, red), (1, white), (2, red), (
2, white)}.
A CARTESIAN PRODUCT is the set of all

ordered pairs from the elements of both sets.
Denoted by ×.
{1, 2, green} × {red, white, green} =
{(1, red), (1, white), (1, green), (2, re
d), (2, white), (2, green), (green, red

17. Cartesian Products
The trick behind CARTESIAN PRODUCTS is

to list down all possible pairs of elements such
that the first element is from the first set and
the second element is from the second set.
A × ∅ = ∅. Zero Property of Products.
1.
A × (B ∪ C) = (A × B) ∪ (A × C). Distribution.
2.
(A ∪ B) × C = (A × C) ∪ (B × C). Distribution.
3.
The CARDINALITY of Cartesian Products is

n(A) × n(B), wherein A and B are the given
sets.

18. Quiz Two: Set Operations
For every slide you are given seven
minutes to answer. Don’t cheat or I’ll kick
your ass. Point system varies per question
difficulty.

19. True or False (One-Point Items)
The cardinality of Cartesian Products is n(A)
1.
× n(B), wherein A and B are the given sets.
Uniting equal sets would result in a new set.
2.
(A’)’ = A.
3.
The intersection of two sets is a subset of
4.
both sets.
If Z = {negative numbers} then Z’ =
5.
{nonnegative numbers}.
AA = U.
6.
The intersection of two sets always has a
7.

20. Set-Building (One-Point Items)
Given the following sets:
U = {m, o, n, s, t, e, r, h, u}
S = {h, u, n, t, s, m, e, n}
R = {r, o, u, t, e} V = {r, e, m, o, t, e} H = Ø
Find the following sets:
(S ∩ V)(R ∩ S)
1. VR 5.
(S ∪ V)R (H ∪ S’) ∩ R
2. 6.
R ∩ V’ (V ∩ R ∩ S)’
3. 7.
U’ ∪ (SV)
H’R
4. 8.

21. Analysis (Two-Point Items)
Write the power set of set P = {x, y}. After
that, create a CARTESIAN PRODUCT
between set P and its power set.
Given that W is the set of days in a
week, and M is the set of months in a year.
Give the CARDINALITY of the Cartesian
Products, and the Union of the Sets.