SETS AND THE REAL NUMBER SYSTEM

A Crash Course for Algebra Dummies

How can something so simple be so f*cking complicated? This is like addition and subtraction, just different terms used.

Lesson Two: Set Operations

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.

Set Unions

Two sets can be "added" 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.

Basic Properties of Unions

1. A ∪ B = B ∪ A. Commutative Property of Unions.

2. A ∪ (B ∪ C) = (A ∪ B) ∪ C. Associative Property of Unions.

3. A ⊆ (A ∪ B). Obviously, since A ∪ B contains A.

4. A ∪ A = A. Uniting equal sets would result in the same set.

5. A ∪ ∅ = A. Identity Property of Unions.6. A ⊆ B if and only if A ∪ B = B. In this case,

A = B, meaning A is an improper subset of B.

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.

Basic Properties of Intersections1. A ∩ B = B ∩ A. Commutative Property of

Intersections.2. A ∩ (B ∩ C) = (A ∩ B) ∩ C. Associative

Property of Intersections.3. A ∩ B ⊆ A. True, since A has a part that B has.4. A ∩ A = A. Same elements from both sets.5. A ∩ ∅ = ∅. Zero Property of Intersections.6. A ⊆ B if and only if A ∩ B = A. In this case, A

= B, following from the 3rd property of intersections.

Exercise 1: Unions & IntersectionsGiven 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)

Answers 1: Unions & Intersections

1. F ∪ U = {s, t, r, a, i, g, h, y}

2. C ∪ K = {l, e, s, b, i, a, n}

3. F ∪ U ∪ C = {s, t, r, a, i, g, h, y, l, e, b, n}

4. U ∩ K = Ø5. F ∩ C = {s, a, i}6. F ∩ U ∩ C = {a}7. (F ∩ C) ∪ (U ∪ K) = {s,

a, i, g, y}

8. (C ∪ U) ∩ (F ∩ K) = Ø

F = {s, t, r, a, i, g, h}

U = {g, a, y}

C = {l, e, s, b, i, a, n}

K = Ø or { }

Set Differences

Two sets can be “subtracted" 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}.

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}

A\B {1, 2, 3, 4}\{1, 3, 5}{2, 4}\{5}

A\B = {2, 4}

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}.

Basic Properties of Complements The trick in COMPLEMENTS is to get the

unmentioned elements in a set to form the new one.

1. (A’)’ = A. Involution Property of Unions.2. U’= Ø. All elements are included in the

universal set.3. Under the same premise, Ø’ = U. 4. A ∪ A’ = U. A’ is all members not part of A in

the universal set.5. Under the same premise, A ∩ A’ = Ø. 6. A\A = Ø. Obvious, since these are equal sets.

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 { }Find the following sets:

1. Q \ P2. (E\P)\Q3. (P\Q) ∩

E4. Q’\P

5. (P ∪ Q)’ ∪ E

6. (P\E) ∩ Q7. (P ∪ E ∪

Q)’8. (D ∩ U)’ ∪

Q

Complements and DifferencesU

= {

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 o

r {

}

1. Q \ P = {f, u, k}2. (E\P)\Q = {m, e}3. (P\Q) ∩ E ={t, h}4. Q’\P ={m, e, r}5. (P ∪ Q)’ ∪ E = {m, e, t,

h} or E6. (P\E) ∩ Q = {c}7. (P ∪ E ∪ Q)’ = Ø8. (D ∩ U)’ ∪ Q = {m, o,

t, h, f, u, c, k, e, r} or U

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, red), (2, white), (2, green), (green, red), (green, white), (green,

green)}.

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.

1. A × ∅ = ∅. Zero Property of Products.2. A × (B ∪ C) = (A × B) ∪ (A × C). Distribution.3. (A ∪ B) × C = (A × C) ∪ (B × C). Distribution. The CARDINALITY of Cartesian Products is

n(A) × n(B), wherein A and B are the given sets.

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.

Quiz Two: Set Operations

True or False (One-Point Items)1. The cardinality of Cartesian Products is n(A)

× n(B), wherein A and B are the given sets. 2. Uniting equal sets would result in a new set.3. (A’)’ = A.4. The intersection of two sets is a subset of

both sets.5. If Z = {negative numbers} then Z’ =

{nonnegative numbers}.6. A\A = U.7. The intersection of two sets always has a

non-zero cardinality.

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:1. V \ R2. (S ∪ V)\

R3. R ∩ V’4. H’\R

5. (S ∩ V)\(R ∩ S)

6. (H ∪ S’) ∩ R7. (V ∩ R ∩ S)’8. U’ ∪ (S\V)

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.