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.
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