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CPSC 121: Models of Computation2008/9 Winter Term 2

Sets

Steve Wolfman, based on notes by Patrice Belleville and others

Lecture Prerequisites

Read Section 5.1.

Solve problems like Exercise Set 5.1, #1-21.

(Note: the notion of a “partition” is important in CS, but we won’t discuss it in CPSC 121. We’ll talk more about “power sets” very soon.)

Complete the open-book, untimed quiz on WebCT that will be due Tuesday evening March 17th.

Learning Goals: Pre-Class

By the start of class, you should be able to:– Define the set operations union, intersection, complement,

and set difference and the logical operations subset and set equality in terms of predicate logic and set membership ().

– Translate between sets represented explicitly (possibly using ellipses “…”, e.g., {4, 6, 8, …}) and using “set builder” notation (e.g., {x Z+ | x2 > 10 x is even}).

– Execute the union, intersection, complement, set difference, subset, and set equality operations on sets expressed explicitly, using set builder notation, or a combination of these and set operators.

– Interpret the empty set symbol , including the fact that the empty set has no members and that it is a subset of any set.

Learning Goals: In-Class

By the end of this unit, you should be able to:– Define the power set and cartesian product

operations in terms of predicate logic and set membership/subset relations.

– Execute the power set, cartesian product, and cardinality operations on sets expressed through any of the notations discussed so far.

– Apply your proof skills to proofs involving sets.– Relate DFAs to sets.

Quiz 10 Notes

Generally excellent results. Only the subset questions caused many problems, because of the empty set.

Set A is a subset of set B exactly when every element of A is an element of B. Is the empty set a subset of B?

Outline

• Out-of-class notes on set definitions• What’s the Use of Sets

– Historical note– Application to DFAs

• More set operations– Cardinality (size)– Power set (and an induction proof)– Cartesian products (and application to DFAs)

• Set proofs

What Are Sets Good For?

Historically:

Mathematicians attempted to formalize set theory to create a foundation for all of mathematics. Essentially all mathematical constructs can be defined in terms of sets.

Good news: this means sets are a powerful way to communicate many types of ideas.

Bad News: Russell’s Paradox(and other problems)

What Are Sets Good For?

Applications for us:

Codifying and communicating ideas. For example, formalizing DFAs...

What is a DFA?

The states are: ______

The start state is: ____

The accepting states are: _____________

The input language is: ________________

What’s left?

a

b

b

a,b

a,ba

The Transition Function

The arrows in our DFA have the following properties:

• An arrow starts from every state, for every letter in the input alphabet.

• The arrows lead to other states.

We’ll formalize this as the function soon, which will complete our DFA!

a

b

b

a,b

a,ba

What is a DFA?(Deterministic Finite Automaton)Q is the (finite) set of

states.

q0 is the start state, and q0 Q.

F is the set of accepting states, and F Q.

Is the (finite) set of letters in the input language.

is the transition function, to be described.

These are just standard names we use,but the constraints make sense regardless of names.

a

b

b

a,b

a,ba

Problem: Formalizing an Example DFA

For this DFA:

Q = { }

q0 =

F = { }

= { }

: ( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

a

b

b

a,b

a,ba

Problem: Testing the DFA Formalism

Q is the set of states

q0 is the start state, and q0 Q.

F is the set of accepting states, and F Q.

Is the set of letters in the input language.

is the transition function, to be described.

Must a DFA have a start state?

Can a DFA have more than one start state?

Must a DFA have an accepting state?

Can all states in a DFA be accepting?

Can the start state be accepting?

Testing the DFA Formalism

Must a DFA have a start state?

Can a DFA have more than one start state?

Must a DFA have an accepting state?

Can all states in a DFA be accepting?

Can the start state be accepting?

q0 is the start state; so, yes.

q0 is the only start state; so, no.

F Q. Is Q?

F Q. Is Q Q?

q0 Q and F Q. Can q0 F?

Must a DFA Have an Accepting State?

What does a DFA with no accepting states look like? What language does it accept?

Can All States in a DFA Be Accepting?

What does a DFA with no rejecting states look like? What language does it accept?

Outline

• Out-of-class notes on set definitions• What’s the Use of Sets

– Historical note– Application to DFAs

• More set operations– Cardinality (size)– Power set (and an induction proof)– Cartesian products (and application to DFAs)

• Set proofs

How Big Is a Set?

How big are these sets?

A = {1, 3}B = {1, 3, 27, snow}C = { {1, 3}, {1, 3, 27, snow} }D = E = NF = { …, -4, -2, 0, 2, 4, … }

Cardinality

The cardinality of a set A — denoted |A| — is the set’s size.

For finite sets, the cardinality of the set is the number of elements it contains…

|A| = |{1, 3}| = 2|B| = |{1, 3, 27, snow}| = 4|C| = |{ {1, 3}, {1, 3, 27, snow} }| = ??|D| = || = ??

Cardinality of Infinite Sets?

For now, we won’t worry about the cardinality of infinite sets. Why?

Well, we can’t count the members of such sets.

Side note: Depending on our definition, however, the results can be surprising. For example, under the standard definition of cardinality, these have the same cardinality:E = N = {1, 2, 3, … }F = { …, -4, -2, 0, 2, 4, … }

Why? Try “folding” the set F over creatively.

Problem: Cardinality Exercises

Given the definitions:

A = {1, 2, 3}

B = {2, 4, 6, 8}

What are:

|A| = _________

|B| = _________

|A B| = _________

|A B| = _________

|A – B| = _________

|B – A| = _________

|{{}}| = _________

|{}| = _________

|{{}}| = _________

Outline

• Out-of-class notes on set definitions• What’s the Use of Sets

– Historical note– Application to DFAs

• More set operations– Cardinality (size)– Power set (and an induction proof)– Cartesian products (and application to DFAs)

• Set proofs

A Touch of Logic

Consider the four propositional logic variables p, q, r, and s.

What are all the different ways can we assign truth values to each of these?

Now, imagine a set S that contains exactly the true variables among p, q, r, and s.

What are the possible “Ss” (depending on p, q, r, and s’s truth values)?

A Touch of Logic: Rephrased

Consider the set {p, q, r, s}.

What are all the subsets of this set?

Equivalently, what is: {S U | S {p,q,r,s}}?

Power Sets

The power set of a set T — denoted P(T) — is the set of all subsets of T:{ S U | S T }

Using Logic to Build Power Sets

p q r s Set from P(…)

F F F F {}

F F F T {s}

F F T F {r}

F F T T {r,s}

F T F F {q}

F T F T {q,s}

F T T F {q,r}

F T T T {q,r,s}

p q r s Set from P(…)

T F F F {p}

T F F T {p,s}

T F T F {p,r}

T F T T {p,r,s}

T T F F {p,q}

T T F T {p,q,s}

T T T F {p,q,r}

T T T T {p,q,r,s}

So, how big is the power set of a set?

P({p,q,r,s}) = …

Problem: Power Set Exercises

Given the definitions: A = {1, 2} B = {2, 4, 6}

What are: P(A) = _____________________________ P(B) = _____________________________ |P(A)| = _____________________________ |P(B)| = _____________________________ |P(P(A))|= _____________________________ P(P(A)) = _____________________________ P(A B) = _____________________________

Cardinality of a Finite Power Set

Theorem: |P(S)| = 2|S|

Base case: When |S| = 0, what is S?

What is P(S)? How do we know?

Cardinality of a Finite Power Set

Induction Hypothesis: Assume for all sets S of cardinality n 0, |P(S)| = 2|S|.

Inductive step:

To prove: Given the induction hypothesis, for all sets T, |P(T)| = 2|T| when |T| = n+1.

The key is to drop an element from a set of size n+1.Then, take the power set.

Then, consider how to put the element back in.

Outline

• Out-of-class notes on set definitions• What’s the Use of Sets

– Historical note– Application to DFAs

• More set operations– Cardinality (size)– Power set (and an induction proof)– Cartesian products (and application to DFAs)

• Set proofs

Tuples

A tuple is an ordered collection of elements. (An n-tuple is a tuple with n elements.)

Two tuples are equal when each pair of corresponding elements is the same:

(a, 1, ) = (a, 5 – 4, A A)(a, b, c) (a, c, b)

Actual fact: database people love the word “tuple”.And… why shouldn’t they? Say “two-tuple”. Isn’t that great?

Formalizing DFAs: Where the Arrows Come From

The arrows in our DFA have the following properties:

• An arrow starts from every state, for every letter in the input alphabet.

• The arrows lead to other states.

We’ll formalize this as a function soon.

What is the set of “things” for which there is an arrow?

a

b

b

a,b

a,ba

t

m

b

g

Cartesian Products

The Cartesian product of the sets A and B — denoted A B — is the set of all tuples whose first element is drawn from A and whose second element is drawn from B.

In other words, A B = {(a,b) | a A b B}

Cartesian Products andthe Cartesian Plane

Let’s visualize N N:

1

2

3

4

5

1 2 3 4 5 6 7 8

Calculating Cartesian Products

What is {1,2,3} {a,b}:

1

2

3

a b

( , )

( , )

( , )

( , )

( , )

( , )

Formalizing DFAs: Where the Arrows Come From

The arrows in our DFA have the following properties:

• An arrow starts from every state, for every letter in the input alphabet.

• The arrows lead to other states.

We’ll formalize this as a function soon.

Each tuple of a state and a letter has an arrow.There’s an arrow for every member of Q

a

b

b

a,b

a,ba

t

m

b

g

Example DFA, Revisited

For this DFA:

Q = { t, m, b, g}

q0 = t

F = { t, b }

= { a, b }

: (t, a) m

(t, b) b

(m, a) g

(m, b) b

(b, a) b

(b, b) b

(g, a) g

(g, b) g

a

b

b

a,b

a,ba

t

m

b

g

Outline

• Out-of-class notes on set definitions• What’s the Use of Sets

– Historical note– Application to DFAs

• More set operations– Cardinality (size)– Power set (and an induction proof)– Cartesian products (and application to DFAs)

• Set proofs

Problem: Proofs with Sets (1/2)

Prove by contradiction that for all sets A and B, if A B then A B = { }.

(Note: B contains all elements in the universe that are not in B.)

Problem: Proofs with Sets (2/2)

Prove for all sets A and B, if A B then P(A) P(B).

(Note: if A B and B C, then A C.)

Learning Goals: In-Class

By the end of this unit, you should be able to:– Define the power set and cartesian product

operations in terms of predicate logic and set membership/subset relations.

– Execute the power set, cartesian product, and cardinality operations on sets expressed through any of the notations discussed so far.

– Apply your proof skills to proofs involving sets.– Relate DFAs to sets.