Discrete Mathematical Structures - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CS 173:

Discrete Mathematical Structures

Set Theory - Definitions and notation

A ⊆ B means “A is a subset of B.”

A ⊇ B means “A is a superset of B.”

A = B if and only if A and B have

exactly the same elements.iff, A ⊆ B and B ⊆ A

iff, A ⊆ B and A ⊇ B iff, ∀x ((x ∈ A) ↔ (x ∈ B)).

So to show equality of sets A and B, show:

• A ⊆ B
• B ⊆ A

Set Theory - Definitions and notation

Quick examples:

Is ∅ ⊆ {1,2,3}?

Yes! ∀x (x ∈ ∅) → (x ∈ {1,2,3}) holds, because (x ∈ ∅) is false.

Is ∅ ∈ {1,2,3}? No!

Is ∅ ⊆ {∅,1,2,3}? Yes!

Is ∅ ∈ {∅,1,2,3}? Yes!

Vacuously

Set Theory - Definitions and notation

Quiz time:

Is {x} ⊆ {x}?

Is {x} ∈ {x,{x}}?

Is {x} ⊆ {x,{x}}?

Is {x} ∈ {x}?

Yes

Yes

Yes

No

Set Theory - Russell’s Paradox

Can we use any predicate P to define a

set

S = { x : P(x) }?

Define S = { x : x is a set where x ∉ x }

Then, if S ∈ S, then by defn of S, S ∉ S. So S must not be in S, right?

But, if S ∉ S, then by defn of S, S ∈ S. ARRRGH!

No!

There is a town with a barber who shaves all the people (and only the people) who don’t shave themselves. Who shaves the barber? Docsity.com

Set Theory - Cardinality

If S is finite, then the cardinality of S,

|S|, is the number of distinct elements

If S = {1,2,3},in S. |S| = 3.

If S = {3,3,3,3,3},

If S = ∅,

If S = { ∅, {∅}, {∅,{∅}} },

|S| = 1.

|S| = 0.

|S| = 3.

If S = {0,1,2,3,…}, |S| is infinite. (more on this later)

Set Theory - Cartesian Product

The Cartesian Product of two sets A and

B is:

If A = {Charlie, Lucy, Linus}, and A x B = { <a,b> : a ∈ A ∧ b ∈ B}

B = {Brown, VanPelt}, then

A,B finite → |AxB| =?

A 1 x A 2 x … x An = {<a 1 , a 2 ,…, an >: a 1 ∈ A 1 , a 2 ∈ A 2 , …,

an ∈ An }

A x B = {<Charlie, Brown>, <Lucy, Brown>,

<Linus, Brown>, <Charlie, VanPelt>, <Lucy, VanPelt>, <Linus, VanPelt>}

We’ll use these special sets soon!

a) AxB b) |A|+|B| c) |A+B| d) |A||B|

Set Theory - Operators

The union of two sets A and B is:

A ∪ B = { x : x ∈ A v x ∈ B}

If A = {Charlie, Lucy, Linus}, and

B = {Lucy, Desi}, then

A ∪ B = {Charlie, Lucy, Linus, Desi}

B A

Set Theory - Operators

The intersection of two sets A and B is:

A ∩ B = { x : x ∈ A ∧ x ∈ B}

If A = {x : x is a US president}, and

B = {x : x is deceased}, then

A ∩ B = {x : x is a deceased US president}

B A

Set Theory - Operators

The intersection of two sets A and B is:

A ∩ B = { x : x ∈ A ∧ x ∈ B}

If A = {x : x is a US president}, and

B = {x : x is in this room}, then

A ∩ B = {x : x is a US president in this room} = ∅

B A

Sets whose intersection is empty are called

disjoint sets

Set Theory - Operators

The set difference, A - B, is:

A

U

B

A - B = { x : x ∈ A ∧ x ∉ B }

A - B = A ∩ B

Set Theory - Operators

The symmetric difference, A ⊕ B, is:

A ⊕ B = { x : (x ∈ A ∧ x ∉ B) v (x ∈ B ∧ x

∉ A)}

= (A - B) U (B - A)

like “exclusive or”

A

U

B

Set Theory - Famous Identities

• Two pages of (almost) obvious.
• One page of HS algebra.
• One page of new.

Don’t memorize them, understand them!

They’re in Rosen, p89.

Set Theory - Famous Identities

• Identity

• Domination

• Idempotent

A ∩ U = A

A U ∅ = A

A U U = U

A ∩ ∅ = A

A U A = A

A ∩ A = A

(Lazy)