Discrete Structures

# Lecture Notes, 25 Oct 2010

 Last Update: 25 October 2010 Note: or material is highlighted

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### §2.1: Sets

1. Where We've Been:

1. I've taught you a language (FOL),

• which we'll "speak" from now on.

2. Now we need something to talk about.

2. Logicism:

1. Many mathematicians, logicians, and philosophers believe that:

• Math = logic + set theory

2. i.e., "sets" are the basic data type of mathematical objects
3. All other mathematical objects can be constructed from,
—i.e., defined in terms of—
sets

4. For a graphic history of logicism, see:

Doxiadis, Apostolos; Papadimitriou, Christos H.; Papadatos, Alecos; & Di Donna, Annie (2009), Logicomix (Bloomsbury USA).

3. What is a set?

1. Def: A set isdef … ?

∃ 2 possible ways to continue this definition:

1. Take "object" as primitive & undefined

• Then define "set" as a "collection" of objects.

• But what's a "collection"?

• This was Cantor's definition, but …

• … it led to Russell's paradox:

R = {x | xx}

2. Take "set" & "member" as primitive & undefined

• But give axioms for them.

2. So: ∃ 2 primitive, undefined types of things:

1. sets
2. members (or: elements)

& one 2-place relation between them ("set membership"):

e ∈ S

3. Notation:

1. "e ∉ S" for: ¬(e ∈ S)
2. "S = {e1, e2, … , en}" for: the set S such that (hereafter: "s.t.") ∀x[x ∈ S ↔ (x=e1) ∨ … ∨ (x=en)]

• This is called an "extensional" description of S;
it shows you all its members.

3. A special case of an extensional description:
{} or ∅ for: the empty set

• i.e., the set S s.t. ¬∃x[x ∈ S]

4. An "intensional" description of S;
it describes all its members:

{x | P(x)} for: the set of all x (in the domain) s.t. P(x)

• i.e., the set S s.t. ∀x[x ∈ S ↔ P(x)]

• Note: This gets us in trouble if we let P(x) = xx:

• See Rosen, p. 121, #38.
• Bertrand Russell's way out:
sets come in "types";
every set must be of a "higher" type than its members.
(So P(x) is simply ungrammatical.)

• E.g.: The set S consisting of all natural numbers n ≤ 4:

0 ∈ S, 1 ∈ S, 2 ∈ S, 3 ∈ S, 4 ∈ S

S = {0,1,2,3,4} (extensional description)
= {x | (xN) ^ (x ≤ 4)} (intensional description)

• E.g.: The set USP of all US presidents:

USP = {GW, JA, TJ, …, GWB, BO, …} (extensionally)
= {x | POTUS(x)} (intensionally)
(where "POTUS(x)" = x is President Of The US)

4. Def: Set Equality:

Let A,B be sets.
Then A = B =def ∀x[x ∈ A ↔ x ∈ B]

1. I.e., a set is "determined by" its members.
2. ∴ teams and clubs are not sets!

3. Sometimes this is presented as an axiom rather than a definition:

• Axiom of Extensionality:
A = B ↔ ∀x[x ∈ A ↔ x ∈ B]

4. E.g.: {1,2,3} = {3,1,2} = {1,1,1,2,2,3,3,3,3}

5. Def: Subset

Let A,B be sets.
Then A is a subset of B =def ∀x[x ∈ A → x ∈ B]

1. I.e., all A's are Bs.

2. Notation: A ⊆ B

3. E.g.: Recall that:
W = the whole numbers = {1,2,3,…}
N = the natural numbers = {0,1,2,3,…}
N+ = the positive natural numbers = {1,2,3,…}
Z = the integers = {…–3,–2,–1,0,1,2,3,…}
Q = the rational numbers
R = the real numbers
C = the complex numbers

WN+NZQRC.

4. Thm:
(∀ set S)[(∅ ⊆ S) ∧ (S ⊆ S)]

• Proof of (∅ ⊆ S) is in text.
• Proof of (S ⊆ S):
Show (S ⊆ S):
Show ∀x[x ∈ S → x ∈ S]:
Choose arb. e ∈ S, & show e ∈ S → e ∈ S:
Supp. e ∈ S, & show e ∈ S:
Trivial! QED

5. Def: Proper Subset

Let A,B be sets.
Then A is a proper subset of B =def ∀x[x ∈ A → x ∈ B] ∧ ∃x[x ∈ B ∧ x ∉ A]

1. Notation: A ⊂ B
2. ∴ A ⊂ B ↔ A ⊆ B ∧ A ≠ B
3. Also: A ⊄ A
4. E.g.: N+NZQRC

6. Def: Power Set

Let S be a set.
Then the power set of S =def {A | A ⊆ S}

1. Notation: ℘(S)

2. E.g.: Let S = {0,1,2}.
Then ℘(S) = {∅, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, S}

Notes:

• {0} ⊆ S, and 0 &isin S.
These are related, but distinct, facts.

• {0} &isin ℘(S), but 0 ∉ ℘(S)

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