CS173 Lecture B

CS173 Lecture B

Tandy Warnow

December 1, 2015

CS 173, Lecture BReview of Cardinality and CountabilityTandy Warnow

Today

I The final exam

I Cardinality of infinite sets (review)

I Countability and how to prove that a set is countable (review)

I Uncountability, and how to prove that a set is not countable(review)

I ICES forms (need a student volunteer to collect and returnthem)

What won’t be on the final exam

What won’t be on the exam:

I Dynamic programming

I The graph algorithm for 2-SAT

I Genome assembly and de Bruijn graphs

The final exam will contain problems on everything else.

I Some of the exam requires proofs (by induction, bycontradiction, etc.).

I Some of it will be multiple choice or True/False.

I You will need to know definitions.

I Nearly all the problems will be verbatim (or almost verbatim)from the exam review document.

So just make sure you can do what’s on the exam reviewdocument!

Examlet on Thursday, December 3

It will have three problems:

1. One proof that a set is countable or perhaps uncountable

2. One problem involving proving a theorem about trees

3. One problem where you are given an adjacency matrix andasked to draw the graph it defines, and then find variousthings. Examples of those things might include:

I Breadth First Search tree starting at a particular nodeI minimum dominating setI minimum vertex coverI maximum matchingI maximum independent setI maximum cliqueI minimum vertex coloring,I Hamiltonian path (if it exists)I Eulerian circuit (if it exists)

Material about Countability/Uncountability

Definitions:

I Countable and uncountable

I Finite and infinite

Techniques:

I Cantor’s diagonalization (proves uncountability)

I Zig-zag (proves countability)

Theorems:

I The countable union of countable sets is countable (zig-zagtechnique).

I The finite product of countable sets is countable (zig-zagtechnique for two sets, induction to generalize).

I Any subset of a countable set is countable.

I Any superset of an uncountable set is uncountable.

Cardinality

The cardinality of a finite set X is the number of elements in X ,and is denoted |X |.Hence, |{1, 2, 3, 4, 5}| = |{2, 9, 12, 17, 18}|.A set X is finite if |X | = n for some n ∈ Z .A set X is infinite if there does not exist any n ∈ Z so that|X | = n.

Note, an alternative definition for infinite is that ∃Y ⊂ X and abijection f : X → Y .How do we talk about cardinality of infinite sets?When can we say that |X | = |Y | for infinite sets |X | and |Y |?

Cardinality of infinite sets

We will say that |X | = |Y | if ∃f : X → Y where f is a bijection.

A set X is countably infinite if there is a bijection from X to N.

Using the prior notation, we say X is countably infinite if |X | = |N|.

In the textbook, the definition of countable is given as “finite orcountably infinite.”

But some authors use countable only for countably infinite... beaware of this.

Proving a set is countable

There are several techniques to prove a set S is countable.

1. Provide a bijection between S and N2. Provide injections (1-1 functions) in each direction between S

and NOr, equally validly, describe an algorithm that produces anenumeration of the elements of S .

Proving that Z− is countable

To prove that Z− = {−1,−2,−3, . . .} is countable, we define abijection f : Z− → N.

I Let f (n) = |n| − 1. We prove that f is a bijection between Z−

and N.I (f ix 1− 1:) Suppose f (x) = f (y). Then |x | − 1 = |y | − 1,

and so |x | = |y |. Since {x , y} ⊂ Z−, both x and y arenegative. Hence x = y , and so f is 1− 1.

I (f is onto:) Let n ∈ N; hence n ≥ 0. Let n′ = −(n + 1). Notethat f (n′) = | − (n + 1)| − 1 = n + 1− 1 = n. Hence f is onto.

Hence Z− is countable.

Proving that Z is countable

Here we show that Z is countable. We have three options:

1. Provide a bijection between Z and N2. Provide injections (1-1 functions) in each direction between Z

and N3. Or, equally validly, describe an algorithm that produces an

enumeration of the elements of Z.

Let’s do the third one.

We will alternate between listing the non-negative integers (N) andthe negative integers (Z−).

So:

0,−1, 1,−2, 2,−3, 3, . . .

This is a perfectly valid proof that Z is countable!

The Zig-Zag Technique

Suppose A and B are both countable. How do we prove thatA ∪ B is countable?

Zig-zag technique:

I Step 1: List the elements of A and B.

I Step 2: List the elements of A ∪ B by alternating between theelements of A and B.

Because A is countable, we can list its elements,A = {a0, a1, a2, . . . , }. Ditto for B, so B = {b0, b1, b2, . . . , }.

Enumerate A ∪ B by alternating between elements of A and B:A ∪ B = {a0, b0, a1, b1, a2, b2, . . .}.

Finite unions of countable sets are countableSuppose A1, A2, . . . ,Ak are all countable. How do we prove that∪k

i=1Ai is countable?

Enumerate the elements of the finite union as follows:I List the first element from each of the k setsI Then list the second element from each of the k setsI Then list the third element from each of the k setsI and so on

Let Ai = {ai0, ai1, ai2, . . . , }, i = 1, 2, . . . , k .

We obtain the list

∪ki=1Ai = {a10, a20, . . . , ak0, a11, a21, . . . , ak1, . . .}

Countable unions of countable sets are countableWe will use the zig-zag technique to prove that ∪i∈NAi iscountable, if each Ai is countable.

Consider the infinite by infinite matrix M, where A0 is in the firstrow, A1 is in the second row, etc.

The i th row lists the elements of Ai .

We need to enumerate all the elements of ∪iAi .

Countable unions of countable sets are countable

Each Ai is countable, i ∈ N. We list all the elements of Ai as thei th row in an infinite by infinite matrix, M.

We then use the zig-zag technique to enumerate all the elements,visiting them as follows:

I M[0, 0]

I M[1, 0], M[0, 1]

I M[2, 0], M[1, 1], M[0, 2]

I etc.

Note that because some elements may appear in more than oneset, we may list an element more than once. We’ll just skip theelement if it’s already appeared in our list.

Countable unions of finite sets are countable

Suppose Ai is a finite set for i = 0, 1, 2, . . .

To prove that ∪iAi is countable, we can just list each set in theorder we see it.Hence, we begin with A0, then we list A1, then we list A2, etc.

The product of two countable sets is countableSuppose A and B are countable. Is A× B countable?

Use the zig-zag technique.

Let A = {a0, a1, . . . , } and B = {b0, b1, . . . , }.

Define matrix M[i , j ] = (ai , bj).

Enumerate the elements of A× B using the zig-zag technique.

The product of finitely many countable sets is countable

Suppose Ai is countable for each i = 0, 1, 2, . . . , k . We prove∏ki=0 Ai countable by induction on k .

I Base case: k = 2 (see previous slide)

I Inductive hypothesis: ∃K ∈ Z+ such that ∀k , 1 ≤ k ≤ K ,∏ki=1 Ai is countable if each Ai is countable.

I By definition,∏K+1

i=0 Ai =∏K

i=0 Ai × AK+1.

I By the inductive hypothesis,∏K

i=0 Ai is countable.

I By the inductive hypothesis, the product of two countable setsis countable.

I Hence∏K+1

i=0 Ai =∏K

i=0 Ai × AK+1 is countable.

So far

We have shown we can prove countability of a set S

I by demonstrating a bijection between S and NI by demonstrating 1-1 functions between S and N in both

directions

I by defining an infinite two-dimensional matrix containing allthe elements of the set, and using a zig-zag technique to listthe elements

Hence:

I The countable union of countable sets is countable

I The finite product of countable sets is countable

I If X ⊆ S and S is countable, then so is X

I If |X | ≤ |S | and S is countable, then so is X

What about uncountability?

A set X is uncountable if X is infinite but |X | 6= |N|.Examples:

I [0, 1]

I RI P(N)

I The set of functions from N to {0, 1}

Furthermore, for any set A that is listed above, then

I Any set X that contains A as a subset is uncountable

I Any set X that contains a subset Y where |Y | = |A| isuncountable

Why P(N) is uncountableThe proof that P(N) is uncountable is in the book, and we’ve goneover it already.

Why [0, 1] is uncountableProof by contradiction.Suppose it is countable. Then we can list the elements asa0, a1, a2, . . ..

Write out the matrix, with ai the i th row, and give each number indecimal notation, written as 0.y (for some y). Note that this ispossible, even for x=1! (In other words, 1.0 = 0.9̄)

Why [0, 1] is uncountable

Hence, if a0 = 0.1701, then the first row would look like:

1, 7, 0, 1, 0, 0, 0, . . .

If a1 = 13 = 0.3̄ then the second row would be:

3, 3, 3, 3, 3, 3, 3, . . .

if a2 = 1, we list it as

9, 9, 9, 9, 9, 9, 9, . . .

We have to define a real number z that is not in this matrix. Howdo we do this?

Why [0, 1] is not countableWe set the digits for z by setting zi so that it is different fromevery element in the matrix in the i th position.

We set be zi = M[i , i ] + 2(mod10).

Why [0, 1] is not countable

The mapping we used is:

I If M[i , i ] = 0 we set zi = 2










Finishing the proof

Now we derive the contradiction!

I We assumed that the set [0, 1] is countable, and that matrixM has a row for every element in the set.

I We defined the number z based on its decimal expansion, sothat zi 6= M[i , i ] for any i .

I Therefore the matrix M cannot have a row for every elementof [0, 1].

I Hence we derive a contradiction, and [0, 1] must beuncountable.

Why didn’t we use zi = M[i , i ] + 1(mod10)?

Note how we defined z ... our mapping...Suppose we had set zi = M[i , i ] + 1(mod10).

Note that .1 = .09̄ (they are really the same number).

Suppose we had obtained z = .1 because the diagonal elementswere 0, 9, 9, 9, 9, . . ..

The first row could have been for the element 0.09̄.

Proving a set is uncountable

To prove a set X is uncountable, do one of the following:

I The same kind of proof by contradiction – enumeration anddiagonalization

I Prove that |X | = |Y | where Y is uncountable

I Find an uncountable set Y and show that Y ⊂ X

I Find an uncountable set Y and a 1-1 function from Y to X ;this is denoted by |Y | ≤ |X |

Other uncountable sets

Using the diagonalization argument, or using other theorems, orcombining these techniques with proofs by contradiction, we cannow prove that the following sets are uncountable:

I P(Y ), where Y is any infinite set

I The set of functions from A to X , where A is countable and|X | ≥ 2 (e.g., A = Z and X = {1, 2, 3}).

I RI R \Q

Summary

Definitions:

I Countable and uncountable

I Finite and infinite

Techniques:

I Cantor’s diagonalization (proves uncountability)

I Zig-zag (proves countability)

Theorems:

I The countable union of countable sets is countable (zig-zagtechnique).

I The finite product of countable sets is countable (zig-zagtechnique for two sets, induction to generalize).

I Any subset of a countable set is countable.

I Any superset of an uncountable set is uncountable.

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CS173 Lecture B