week 2 tutorial 2.pdf · 2020-04-16 · tutorial format we’ll be walking through some problems...

Week 2 TutorialSet Theory and Proofwriting

General LogisticsAs a quick reminder:

– Tutorials are 1 hour sessions every week

– We encourage you to try and choose the same session week to week, but this is not required

– We will record one session per week – if you’re unable to attend any tutorials, you may make up the exercises by Thursday noon PDT

– You must attend or make up 7/10 tutorials

Tutorial Format● We’ll be walking through some problems

designed to solidify the concepts covered in this week’s assignment.

● These will focus on problem-solving techniques rather than teaching new content, so the expectation is that you’re caught up on lectures!

● We’ll periodically split of into breakout rooms, where you’ll get a chance to discuss in smaller groups.

Tutorial Exercises● Each tutorial has a corresponding

assignment in Gradescope consisting of a few short answer questions.

● During the live tutorial sessions, we’ll complete these questions together.

● If you are making up a tutorial, you will be responsible for watching the recording and submitting answers for the exercises on your own.

Introduction:How to Approach CS103

● “Everyone else has been doing math since before they were born and there is no way I'll ever be as good as them.”

● “A small minority of people are math geniuses and everyone else has no chance at being good at math.”

● “Being good at math means being able to instantly solve any math problem thrown at you.”

Mental Traps to Avoid

Time

Ab

ilit

y

“A little slope makes up for a lot of y-intercept.” - John Ousterhout

Pro Tip #1:

Never Confuse Experience for Talent

Pro Tip #2:

Have a Growth Mindset

Suppose you improve at some skill at a rate of 1% per day. How much better at that skill

will you be by the end of the year?

After one day, you're 1.01 times better.After two days, you're (1.01)2 times better.

After one year, you'll be (1.01)365 ≈ 37.8 times better!

Fun Math Question

Pro Tip #3:

Avoid an Ingroup/Outgroup Mindset

● Math is often driven by seemingly simple problems that no one knows the answer to.

● Example: the integer brick problem:

Is there a rectangular brick where any line connecting two corners has integer length?

● Having open problems like these drives the feld forward – it motivates people to fnd new discoveries and to invent new techniques.

Simple Open Problems

?

Don't Psych Yourself Out● It is perfectly normal to get stuck or

be confused when learning math.

● We’ve all been on the Struggle Bus. Don’t be afraid to ask for help!

● Engage with the concepts. Work through lots of practice problems. Play around with new terms and defnitions on your own time to see how they work.

● Ask for help when you need it. We're here to help you. We want you to succeed, so let us know what we can do to help!

● Work in groups. Get help from your problem set partner, the TAs, and your tutorial session buddies.

Getting Good at Math

Set Theory Warmup

1. Answer each of the following questions:a) Which pairs of the above sets, if any, are equal to one another?b) Is D ∈ A? Is D ⊆ A?c) What is A ∩ C? How about A ∪ C? How about A Δ C?d) What is A – C? How about {A – C}? Are those sets equal?e) What is |B|? What is |E|? What is |F|?f) What is E – A? Express your answer in set-builder notation.g) Is 0 ∈ E? Is 0 ∈ F?

Fill in answer on Gradescope!

1. Answer each of the following questions:a) Which pairs of the above sets, if any, are equal to one another?b) Is D ∈ A? Is D ⊆ A?c) What is A ∩ C? How about A ∪ C? How about A Δ C?d) What is A – C? How about {A – C}? Are those sets equal?e) What is |B|? What is |E|? What is |F|?f) What is E – A? Express your answer in set-builder notation.g) Is 0 ∈ E? Is 0 ∈ F?


Consider the following sets:

A = { 0, 1, 2, 3, 4 }B = { 2, 2, 2, 1, 4, 0, 3 }C = { 1, {2}, {{3, 4}} }D = { 1, 3 }E = ℕF = { ℕ }

An Incorrect Set Theory Proof

Claim: If A, B, and C are sets and C ⊆ A ∪ B, then C ⊆ A or C ⊆ B.

⚠ Incorrect! ⚠ Proof: Consider arbitrary sets A, B, and C where C ⊆ A ∪ B.

This means that every element of C is in either A or B. If all elements of C are in A, then C ⊆ A. Alternately, if everything in C is in B, then C ⊆ B. In either case, everything inside of C has to be contained in at least one of these sets, so the theorem is true. ■




2. What’s wrong with this proof?


2. What’s wrong with this proof?



This is just repeating defnitions and not making specifc claims about

specifc variables.

This is just repeating defnitions and not making specifc claims about

specifc variables.






Why is this bad?Why is this bad?




While this claim is true, it does not imply the theorem is true. In fact,

this theorem is actually false.

While this claim is true, it does not imply the theorem is true. In fact,

this theorem is actually false.

Let’s Draw Some Pictures!Claim: If A, B, and C are sets and C ⊆ A ∪ B, then C ⊆ A or C ⊆ B.

A


A B


A B

A ∪ B


A B

A ∪ B

Recall the intuition of a subset being “something

I can circle”


I can circle”



I can circle”


I can circle”


A

So C ⊆ A would mean that C is something I

can circle in this region.

So C ⊆ A would mean that C is something I

can circle in this region.

C


I can circle”


I can circle”


Likewise, C ⊆ B would mean that C is

something I can circle in this region.

Likewise, C ⊆ B would mean that C is

something I can circle in this region.

B

C

A ∪ B


A ∪ B

But when I look at A∪B, I can draw C as a circle containing elements from both A and B



C

A ∪ B




Do you see why this circle is in neither A

nor B?

Do you see why this circle is in neither A

nor B? C

A ∪ B


C

3. Using this visual intuition, come up with a counterexample to this claim and

write it up as a disproof. Fill in answer on Gradescope!

3. Using this visual intuition, come up with a counterexample to this claim and

write it up as a disproof. Fill in answer on Gradescope!

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B.


Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}


A

1

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}, B = {2}


A

1

B

2

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}, B = {2}, and C = {1, 2}.


A

1

B

2

C

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}, B = {2}, and C = {1, 2}. Now notice that {1, 2} ⊆ A ∪ B so C ⊆ A ∪ B


A

1

B

2

C

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}, B = {2}, and C = {1, 2}. Now notice that {1, 2} ⊆ A ∪ B so C ⊆ A ∪ B, but C ⊆⊈ A because 2 ∈ C but 2 ∉ A


A

1

B

2

C

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}, B = {2}, and C = {1, 2}. Now notice that {1, 2} ⊆ A ∪ B so C ⊆ A ∪ B, but C ⊆⊈ A because 2 ∈ C but 2 ∉ A, and C ⊆⊈ B because 1 ∈ C but 1 ∉ B.


1

B

2

A

C

Disproof: We will show that there are sets A, B, and Cwhere C ⊆ A ∪ B, but C ⊆⊈ A and C ⊆⊈ B. Consider the sets A = {1}, B = {2}, and C = {1, 2}. Now notice that {1, 2} ⊆ A ∪ B so C ⊆ A ∪ B, but C ⊈⊈ A because 2 ∈ C but 2 ∉ A, and C ⊆⊈ B because 1 ∈ C but 1 ∉ B.

Thus we’ve found a set C which is a subset of A ∪ B but is not a subset of either A or B, which is what we needed to show. ■


Proofwriting Advice

● Be very wary of proofs that speak generally about “all objects” of a particular type.● As you’ve just seen, it’s easy to

accidentally prove a false statement at this level of detail.

● Making broad, high-level claims often indicates deeper logic errors or conceptual misunderstanding (like code smell but for proofs!)

Proofwriting Advice

A Very Good Idea: After you’ve written a draft of a proof, run through all of the points on the Proofwriting Checklist.● This is a great exercise that you can

do with a partner!

Let’s do a proof together!

Theorem: If A, B, and C are sets and C ⊆ A ∩ B, then C ⊆ A and C ⊆ B.

Hold on, isn’t this the claim we just disproved?Hold on, isn’t this the claim we just disproved?


Notice that that’s an intersection, not a union! It turns out that this claim is actually true.

Notice that that’s an intersection, not a union! It turns out that this claim is actually true.


Let’s Draw Some Pictures!Theorem: If A, B, and C are sets and C ⊆ A ∩ B, then C ⊆ A and C ⊆ B.

A


A B


A B

A ∩ B


A B

A ∩ B

Let’s Draw Some Pictures!


I can circle”


I can circle”


A B

A ∩ B

Let’s Draw Some Pictures!


I can circle”


I can circle”


When I look at A∩B, any circle I can draw in this region can be

found in both A and B.

When I look at A∩B, any circle I can draw in this region can be

found in both A and B.

C

A B

A ∩ B


This is a great visual intuition to see why the theorem is true. Now we have to drill down to the

level of individual elements to write the

proof.

This is a great visual intuition to see why the theorem is true. Now we have to drill down to the

level of individual elements to write the

proof.

C

What We’re Assuming What We Need To Show

When confronted with a theorem to prove, the frst step is to make sure you understand where you’re

starting and where you’re going.

When confronted with a theorem to prove, the frst step is to make sure you understand where you’re

starting and where you’re going.



● A, B, and C are sets● C ⊆ A∩B

● C ⊆ A and C ⊆ B






A great proofwriting strategy is to write down relevant definitions. This gives you a better sense of what you need to prove and what tools you

have at hand.

A great proofwriting strategy is to write down relevant definitions. This gives you a better sense of what you need to prove and what tools you

have at hand.





Before we start: - What is the defnition of subset? - How do you prove that one set is a subset of

another? - If you know that one set is a subset of

another, what can you conclude?

Before we start: - What is the defnition of subset? - How do you prove that one set is a subset of

another? - If you know that one set is a subset of

another, what can you conclude?





Definition: If S and T are sets, then S ⊆ T whenfor every x ∈ S, we have x ∈ T.







To prove that S ⊆ T: Pick an arbitrary x ∈ S, then prove x ∈ T.

If you know that S ⊆ T: If you have an x ∈ S, you can conclude x ∈ T.










Brief Interlude: Defnitions in Proofs

● Typically, you will leverage defnitions in your proofwriting in the following two ways:









● Scafolding a proof: You should be using the defnition to identify the start and end points of your proof.









● Applying to known objects: If you know or discover that a defnition holds for some particular object(s), you can establish something about those objects and hopefully make progress towards your goal.











This reading of the defnition is usually

helpful for unpacking this column!







Our Tools

● In general to show that S ⊆ T, pick an arbitrary x ∈ S, show that x ∈ T

In general to show that S ⊆ T, pick an arbitrary x ∈ S, show that x ∈ T





Our Tools



How can we apply this general template to our

specifc problem?

How can we apply this general template to our

specifc problem?



● C ⊆ A and C ⊆ B● Pick an x ∈ C, show that x ∈ A

● Pick an x ∈ C, show that x ∈ B


Our Tools








Our Tools



Now we know that ultimately, we’re going to have to do these two

things. Let’s see what tools we have that can get us here!

Now we know that ultimately, we’re going to have to do these two

things. Let’s see what tools we have that can get us here!






Our Tools








Our Tools



What We Need To Show
















Our Tools



Before we continue: - What is the defnition of set intersection?

Before we continue: - What is the defnition of set intersection?






Our Tools



Definition: The set S ∩ T is the set where, for any x:x ∈ S ∩ T when x ∈ S and x ∈ T







Our Tools




To prove that x ∈ S ∩ T: Prove both that x ∈ S and that x ∈ T.

If you know that x ∈ S ∩ T: You can conclude both that x ∈ S and that x ∈ T.









This is the one we want!

This is the one we want!

Our Tools














Our Tools


If you know that S ⊆ T and you have an x ∈ S, you can conclude x ∈ T.

If you know that x ∈ S∩T, we can conclude that x ∈ S and x ∈ T.w that S ⊆ T, pick an arbitrary x ∈ S






Our Tools


● If you know that S ⊆ T and you have an x ∈ S, you can conclude x ∈ T.

● If you know that x ∈ S∩T, we can conclude that x ∈ S and x ∈ T.w that S ⊆ T, pick an arbitrary x ∈ S






Our Tools



● If you know that x ∈ S∩T, we can conclude that x ∈ S and x ∈ T. that S ⊆ T, pick an arbitrary x ∈ S

Let’s go and try and do the proof with what we’ve

got here!

Let’s go and try and do the proof with what we’ve

got here!

Rough Outline Relevant Definitions

● Assume C ⊆ A ∩ B

Proving C ⊆ A

Pick an x ∈ C

x ∈ A ∩ B

x ∈ A and x ∈ B

Conclude x ∈ A




● If you know that x ∈ S ∩ T, we can conclude that x ∈ S and x ∈ T. that S ⊆ T, pick an arbitrary x ∈ S


● Assume C ⊆ A ∩ B● Proving C ⊆ A

● Pick an x ∈ C

x ∈ A ∩ B

x ∈ A and x ∈ B● Conclude x ∈ A







● Pick an x ∈ C

x ∈ A ∩ B






What goes here?What goes here?



● Pick an x ∈ C● x ∈ A ∩ B








● Pick an x ∈ C● x ∈ A ∩ B● x ∈ A and x ∈ B● Conclude x ∈ A





Rough Outline




Rough Outline




We also need to prove that C ⊆ B.

Notice that if you take the outline here and literally

swap the variable A for the variable B, you get a

working proof.

We also need to prove that C ⊆ B.

Notice that if you take the outline here and literally

swap the variable A for the variable B, you get a

working proof.

Rough Outline

● Assume C ⊆ B ∩ A● Proving C ⊆ B

● Pick an x ∈ C● x ∈ B ∩ A● x ∈ B and x ∈ A● Conclude x ∈ B


In a case like this where your proof would have two

completely symmetric branches, it’s fne to write

up just one and say “by symmetry, [the other branch] is also true.”

In a case like this where your proof would have two

completely symmetric branches, it’s fne to write

up just one and say “by symmetry, [the other branch] is also true.”

Rough Outline




4. Take a few minutes and write up a proof of the theorem using

this outline.

Then share your proof within your group and critique each

other!


4. Take a few minutes and write up a proof of the theorem using

this outline.

Then share your proof within your group and critique each

other!



Proof: Let A, B, and C be arbitrary sets where C ⊆ A ∩ B. We need to show that C ⊆ A and C ⊆ B.

Because the roles of A and B in this proof are symmetric, we can just prove that C ⊆ A.

Choose any element x ∈ C. Since C ⊆ A ∩ B, we know that x ∈ A ∩ B. This tells us that x ∈ A and x ∈ B. In particular, this means that x ∈ A, thus completing the proof. ■





Are you clearly stating what you’re assuming and what you’re trying to prove?

Are you clearly stating what you’re assuming and what you’re trying to prove?




Choose any element x ∈ C. Since C ⊆ A ∩ B, we know that x ∈ A ∩ B. This tells us that x ∈ A and x ∈ B. In particular, this means that x ∈ A, thus completing the proof. ■ Are you making specifc claims about specifc variables? Your proof should NOT have statements of the form “every

element of C”.

Are you making specifc claims about specifc variables? Your proof should NOT have statements of the form “every

element of C”.





Are all variables properly introduced and scoped? You should be able to point at every variable and say that it is either:

1) an arbitrarily chosen value2) an existentially instantiated value

3) an explicitly chosen value

Are all variables properly introduced and scoped? You should be able to point at every variable and say that it is either:

1) an arbitrarily chosen value2) an existentially instantiated value

3) an explicitly chosen value

Proofwriting Strategies

● Articulate a Clear Start and End Point● What are you assuming? What are you trying to

prove?● Write Down Relevant Terms and

Definitions● Identify existing tools to help you get from your

starting point to your ending point● Work Backwards

● Use your end goal to fgure out intermediate steps

week 2 tutorial 2.pdf · 2020-04-16 · tutorial format we’ll be walking through some problems...

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