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CS 173: Discrete Mathematical StructuresPowerPoint Presentation

CS 173: Discrete Mathematical Structures

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CS 173: Discrete Mathematical Structures. Dan Cranston [email protected] Rm 3240 Siebel Center Office Hours: Wed, Fri 11:30a-12:30p. CS 173 Announcements. Homework 2 returned this week (by email, in section, or in your section leader ’ s office hours).

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### CS 173:Discrete Mathematical Structures

Dan Cranston

Rm 3240 Siebel Center

Office Hours: Wed, Fri 11:30a-12:30p

CS 173 Announcements

- Homework 2 returned this week (by email,
in section, or in your section leader’s office hours).

- Homework 3 available. Due 02/04, 8a.

Set Theory - Definitions and notation

A set is an unordered collection of elements.

Some examples:

{1, 2, 3} is the set containing “1” and “2” and “3.”

{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant.

{1, 2, 3} = {3, 2, 1} since sets are unordered.

{1, 2, 3, …} is a way we denote an infinite set (in this case, the natural numbers).

= {} is the empty set, or the set containing no elements.

Venn Diagram

B

Set Theory - Definitions and notationx S means “x is an element of set S.”

x S means “x is not an element of set S.”

A B means “A is a subset of B.”

or, “B contains A.”

or, “every element of A is also in B.”

or, x ((x A) (x B)).

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

B

Set Theory - Definitions and notationA B means “A is a proper subset of B.”

- A B, and A B.
- x ((x A) (x B)) x ((x B) (x A))
- x ((x A) (x B)) x ((x B) (x A))

Set Theory - Definitions and notation

Quick examples:

- {1,2,3} {1,2,3,4,5}
- {1,2,3} {1,2,3,4,5}
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!

Yes

Yes

No

Set Theory - Definitions and notationQuiz time:

Is {x} {x}?

Is {x} {x,{x}}?

Is {x} {x,{x}}?

Is {x} {x}?

: and | are read “such that” or “where”

Primes

Set Theory - Ways to define sets- Explicitly: {John, Paul, George, Ringo}
- Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…}
- Set builder: { x : x is prime }, { x | x is odd }. In general { x : P(x) is true }, where P(x) is some description of the set.

Ex. Let D(x,y) denote “x is divisible by y.”

Give another name for

{ x : y ((y > 1) (y < x)) D(x,y) }.

|S| = 1.

|S| = 0.

|S| = 3.

Set Theory - CardinalityIf S is finite, then the cardinality of S, |S|, is the number of distinct elements in S.

If S = {1,2,3},

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

If S = ,

If S = { 1, {1}, {1,{1}} },

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

We say, “P(S) is the set of all subsets of S.”

2S = {, {a}}.

2S = {, {a}, {b}, {a,b}}.

2S = {}.

2S = {, {}, {{}}, {,{}}}.

Set Theory - Power setsIf S is a set, then the power set of S is

2S = { x : x S }.

If S = {a},

If S = {a,b},

If S = ,

If S = {,{}},

Fact: if S is finite, |P(S)| = 2|S|. (if |S| = n, |P(S)| = 2n)

A,B finite |AxB| = ?

- |A|+|B|
- |A|-|B|
- |A||B|

The Cartesian Product of two sets A and B is:

A x B = { <a,b> : a A b B}

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

B = {Brown, VanPelt}, then

A x B = {<Charlie, Brown>, <Lucy, Brown>, <Linus, Brown>, <Charlie, VanPelt>, <Lucy, VanPelt>, <Linus, VanPelt>}

A1 x A2 x … x An = {<a1, a2,…, an>: a1 A1, a2 A2, …, an An}

A

Set Theory - OperatorsThe 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}

A

Set Theory - OperatorsThe intersection of two sets A and B is:

A B = { x : x A x B}

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

B = {Lucy, Desi}, then

A B = {Lucy}

A

Set Theory - OperatorsAnother example

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

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

A

Sets whose intersection is empty are called disjoint sets

Set Theory - OperatorsOne more example

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} =

- = U and
U =

The complement of a set A is:

A = { x : x A}

If A = {x : x is bored},

then A = {x : x is not bored}

U

A

B

Set Theory - OperatorsThe set difference, A - B, is:

A - B = { x : x A x B }

A - B = A B

A

B

like “exclusive or”

Set Theory - OperatorsThe symmetric difference, A B, is:

A B = { x : (x A x B) v (x B x A)}

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

Set Theory - Operators

A B = { x : (x A x B) v (x B x A)}

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

Proof:

{ x : (x A x B) v (x B x A)}

= { x : (x A - B) v (x B - A)}

= { x : x ((A - B) U (B - A))}

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

Don’t memorize them, understand them!

They’re in Rosen, p124.

Set Theory - Famous Identities- Two pages of (almost) obvious.
- One page of HS algebra.
- One page of new.

A U = A

A U U = U

A U A = A

A U = A

A =

A A = A

Set Theory - Famous Identities- Identity
- Domination
- Idempotent

B A

A U (B U C)

A U (B C) =

A (B C)

A (B U C) =

Set Theory - Famous IdentitiesA U B =

- Commutativity
- Associativity
- Distributivity

A B =

(A U B)U C =

(A B) C =

(A U B) (A U C)

(A B) U (A C)

(A UB)= A B

(A B)= A U B

Hand waving is good for intuition, but we aim for a more formal proof.

Set Theory - Famous Identities- DeMorgan’s I
- DeMorgan’s II

p

q

Like

Not hard, a little tedious

Set Theory - 4 Ways to prove identities- Show that A B and that A B.
- Use a membership table.
- Use previously proven identities.
- Use logical equivalences to prove equivalent set definitions.

(A UB)= A B

Set Theory - 4 Ways to prove identitiesProve that

- () (x A U B) (x A U B) (x A and x B) (x A B)
2. () (x A B) (x A and x B) (x A U B) (x A U B)

(A UB)= A B

Haven’t we seen this before?

Set Theory - 4 Ways to prove identitiesProve that

using a membership table.

0 : x is not in the specified set

1 : otherwise

(A UB)= A B

(A UB)= {x : (x A v x B)}

= A B

= {x : (x A) (x B)}

Set Theory - 4 Ways to prove identitiesProve that using logically equivalent set definitions.

= {x : (x A) (x B)}

Set Theory - A proof for you to do

X (Y - Z) = (X Y) - (X Z). True or False?

Prove your response.

(X Y) - (X Z) = (X Y) (X Z)’

= (X Y) (X’ U Z’)

= (X Y X’) U (X Y Z’)

= U (X Y Z’)

= (X Y Z’)

A U B =

A = B

A B =

A-B = B-A =

Trying to pv p --> q

Assume p and not q, and find a contradiction.

Our contradiction was that sets weren’t equal.

Set Theory - A proof for us to do together.A B =

Prove that if (A - B) U (B - A) = (A U B) then ______

Suppose to the contrary, that A B , and that x A B.

Then x cannot be in A-B and x cannot be in B-A.

So x is not in (A - B) U (B - A).

But x is in A U B since (A B) (A U B).

Thus, A B = .

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