Ch. 2.1: Sets, Subsets, and Venn Diagrams

Ch. 2.1: Sets, Subsets, and Venn Diagrams

A Set may be thought of as a collection of objects, called elements, or members.
Elements can be physical objects, abstract ideas, or even other sets. Here are some
examples...

A = {2, 4, 6, 8, . . . }
C = {$, £, ₩, ¤}
S = {all students attending WSU}
M = {x | x is a student,  x is an art major.}
• A set is usually denoted by a capital letter (usually Roman, sometimes
Greek).

• Elements are usually denoted by a lowercase letter (though some elements
can be sets themselves!).

• The order in which elements are listed between the curly braces does not
matter.

• There should be no repetitions in the list; i.e, all elements should be unique
from each other, in some way.

Set Relations:

Notation... Read... Meaning...

A=B A equals B The sets A and B have the exact same elements.
x∈A x is in A x is an element of set A.
x ∈/ A x is not in A x is NOT an element of set A.
A⊆B A is a subset of B Every element of set A is also an element of set B.
A∩B=∅ A is disjoint from B The sets A and B have no elements in common.

Examples...
Cairo ∈/ { all cities in the United States },
d ∈ {a, b, c, d, r}
{x | x is a college football player} ⊆ {x | x is an athlete}
(in words: ‘‘all college football players are athletes’’)
{ all even numbers } ∩ { all prime numbers } = ∅
Note that: {a, b} ∈ { , , {a, b}, 50}.

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However: {a, b} ⊆ { , , a, b, 50}

• Note that A ⊆ A for any set A.
• The notation A ⊂ B means that ‘‘A is a subset of B, but A = B’’ – we call A

a proper subset of B.
• Sometimes we write A ⊇ B and say that A is a superset of B (or if A ⊃ B,

we can say A is a proper superset of B)

Special Sets:

The Universal Set or Universe of Discourse: U or Ω. The set of everything in
the universe... well, realistically, a set of all possible elements under consideration.
This depends on the context of the discussion: for example, in a game of poker
we may be interested in the set of all playing cards in a deck.

The Empty Set: ∅ or {}. The unique set that contains no elements.
It is always true that ∅ ⊆ A for any set A.

Sets of Numbers:

Natural Numbers N = {1, 2, 3, 4, . . . }
Integers Z = {. . . , −2, −1, 0, 1, 2, 3, . . . }
Rational Numbers Q = {x | x √= m/n, where m, n ∈ Z  n = 0}
Real Numbers R = {e, π, 2, etc...}
Complex Numbers C = {x | x = a + bi, where a, b ∈ R}.

Note that N ⊂ Z ⊂ Q ⊂ R ⊂ C.

Power Set: The set of all possible subsets of a set. Denoted P(A) for some set A.

Example: Let A = {, , }. Then...
P(A) = {∅, {}, {}, {}, {, }, {, }, {, }, {, , }}

Note that ∅ ∈ P(A) and A ∈ P(A). This is always true of power sets.

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Cardinality:

The cardinality of a set, n(A) or |A|, is simply the total number of elements in
that set. For example...

n({1, 2, 3, 4, 5}) = 5
n({3, 4, 5, 6, 7}) =
n({{1, 2}, 3}) = 2
n({{x, y, z}, {0, 9, 8}, {{ 1 , 3 }, {i, j}}}) =

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|{0}| = 1
|∅| = 0
If |B| = 10, and A ⊆ B,
then what can you say about |A| ?
Note: For sets with infinitely many elements, as N and R, cardinality has to be
defined more carefully. You don’t have to worry about that now: it’s beyond the
scope of our course. But just FYI, it turns out that some infinite sets are ‘‘larger’’
than other infinite sets, while some have the same cardinality as a proper subset!
For instance, it turns out that |N| = |Q| and |N| < |R|.

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Ch. 2.2: Operations with Sets:

We can combine or manipulate any set to form a new set, using 3 basic operations:

Notation... Read... Meaning...

A∪B A union B The set of all elements contained in A OR B.
A∩B A intersect B The set of all elements contained in A AND B, simultaneously.
A A-complement The set of all elements NOT in A.

Here are some examples...
Let U = {a, b, c, d, e, f, g, h, i, j} and
S = {a, b, d, g, i},
G = {b, c, e, f, g},
H = {a, d, j}.

S∪G=
S∩G=
S∩G=
G∩H=
(G ∪ H) ∩ S =
G ∪ (H ∩ S) =

If we let U = Z, then {x | x ≥ −2} ∩ {x | x < 5} =

Try to simplify the following expressions:
A∩∅=
A∪A=
(A ∩ B) ∪ (A ∩ B) =

Can you draw a Venn-Diagram for... (A ∩ B) ?
How about... (A ∩ B) ∪ (B ∩ A) ?

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