CSE 311 Lecture 10: Set Theory
Emina Torlak and Kevin Zatloukal
Topics
English proofs and proof strategies
A quick wrap-up of Lecture 09 .
Set theory basics
Set membership ($\in$), subset ($\subseteq$), and equality ($=$).
Set operations
Set operations and their relation to Boolean algebra.
More sets
Power set, Cartesian product, and Russell’s paradox.
Working with sets
Representing sets as bitvectors and applications of bitvectors.
English proofs and proof strategies
A quick wrap-up of Lecture 09 .
Benefits of English proofs
This is more work to write
%a = add %i , 1
%b = mod %a , %n
%c = add %arr , %b
%d = load %c
%e = add %arr , %i
store %e , %d
than this
Higher level language is easier because it skips details.
Formal proofs are the low level language: each part must be spelled out in
precise detail.
English proofs are the high level language.
An English proof is correct if the reader is convinced they can “compile” it to a formal proof if necessary.
Proof strategies
Sometimes, it’s too hard to prove a theorem directly using
inference rules, equivelances, and domain properties.
When that’s the case, try one of the following alternative strategies:
Proof by contrapositive,
Disproof by counterexamples, and
Proof by contradiction.
Proof by contrapositive
If we assume $\neg q$ and derive $\neg p$, then we have proven that
$\neg q \rightarrow \neg p$, which is equivalent to proving $p \rightarrow q$.
1.1.
$\neg q$
Assumption
$\ldots$
1.3.
$\neg p$
2.
$\neg q \rightarrow \neg p$
Direct Proof Rule
3.
$p \rightarrow q$
Contrapositive: 2
Counterexamples
To disprove $\forall x. P(x)$, prove $\exists x. \neg P(x)$.
Works by DeMorgan’s Law: $\neg \forall x. P(x)\equiv \exists x.\neg P(x)$.
All we need to do is find an $x$ for which $P(x)$ is false.
This $x$ is called a counterexample .
Example: disprove that “Every prime number is odd”.
2 is a prime number that is not odd.
Proof by contradiction
If we assume $p$ and derive $\mathsf{F}$ (a contradiction), then we have proven $\neg p$.
1.1.
$p$
Assumption
$\ldots$
1.3.
$\mathsf{F}$
2.
$p \rightarrow \mathsf{F}$
Direct Proof Rule
3.
$\neg p \vee \mathsf{F}$
Law of Implication: 2
4.
$\neg p$
Identity: 3
An example proof by contradiction
Prove that “No integer is both even and odd.”
English proof: $\neg\exists x. \text{Even}(x)\wedge\text{Odd}(x) \equiv \forall x. \neg (\text{Even}(x)\wedge\text{Odd}(x))$.
Proof by contradiction
Let $x$ be an arbitrary integer and suppose that it is both even and odd.
Then $x=2a$ for some integer $a$ and $x=2b+1$ for some integer $b$.
Therefore $2a=2b+1$ and hence $a=b+\frac{1}{2}$.
But two integers cannot differ by $\frac{1}{2}$ so this is a contradiction.
Therefore no integer is both even and odd. $\qed$
Fun strategy: proof by computer
Use an automated theorem prover:
; No integer is both even and odd.
( define-fun even (( x Int )) Bool
( exists (( y Int )) ( = x ( * 2 y ))))
( define-fun odd (( x Int )) Bool
( exists (( y Int )) ( = x ( + ( * 2 y ) 1 ))))
( define-fun claim () Bool
( not ( exists (( x Int )) ( and ( even x ) ( odd x )))))
( assert ( not claim )) ; proof by contradiction
( check-sat )
While this example works, proofs of arbitrary formulas in predicate logic cannot be automated. But interactive theorem provers can still help by checking your formal proof and filling in some low-level details for you.
Fun fact: counterexamples & contradiction in verification
Automated verifiers work by counterexample and contradiction proofs.
Recall that program verification involves proving that a program $P$ satisfies
a specification $S$ on all inputs $x$: $\forall x. p(x) \rightarrow s(x)$, where
$p$ and $s$ are formulas encoding the semantics of $P$ and $S$.
The program verifier sends the formula $\exists x. p(x) \wedge \neg s(x)$ to the prover.
$\neg \forall x. p(x) \rightarrow s(x) \equiv \exists x. \neg (p(x) \rightarrow s(x)) \equiv \exists x. \neg (\neg p(x) \vee s(x))\equiv \exists x. p(x) \wedge \neg s(x)$.
If the prover finds a counterexample, we know the program is incorrect.
The counterexample is a concrete input (test case) on which the program violates the spec.
If no counterexample exists, we know the program is correct.
Because this is proof by contradiction! The prover assumed $\exists x. p(x) \wedge \neg s(x)$ and arrived at false (“unsat”).
Set theory basics
Set membership ($\in$), subset ($\subseteq$), and equality ($=$).
What is a set?
A set is a collection of objects called elements .
Write $a\in B$ to say that $a$ is an element in the set $B$.
Write $a \not\in B$ to say that $a$ isn’t an element of $B$.
Examples
$A = \{1\}$
$B = \{1, 3, 2\}$
$C = \{\triangle, 1\}$
$D = \{\{17\}, 17\}$
$E = \{1, 2, 7, \alpha, \varnothing, \text{dog}, $ $\}$
Some common sets
$\mathbb{N}$ is the set of Natural Numbers : $\mathbb{N}=\{0,1,2,\ldots\}$
$\mathbb{Z}$ is the set of Integers : $\mathbb{Z}=\{\ldots, -2, -1, 0,1,2,\ldots\}$
$\mathbb{Q}$ is the set of Rational Numbers : e.g. $\frac{1}{2}, -17, \frac{32}{48}$
$\mathbb{R}$ is the set of Real Numbers : e.g. $1, -17, \frac{32}{48}, \pi, \sqrt{2}$
$[n]$ is the set $\{1,2,\ldots,n\}$ where $n$ is a natural number.
$\{\}=\varnothing$ is the empty set ; the only set with no elements.
Sets can be elements of other sets
For example, consider the sets
$A = \{\{1\},\{2\},\{1,2\},\varnothing\}$
$B = \{1,2\}$
Then we have
$B \in A$
$\varnothing \in A$
Definitions: equality and subset
$A$ and $B$ are equal if they have the same elements.
$A=B \equiv \forall x. x\in A \leftrightarrow x\in B$
$A$ is a subset of $B$ if every element of $A$ is also in $B$.
$A \subseteq B \equiv \forall x. x\in A \rightarrow x\in B$
Note: $A=B \equiv A \subseteq B \wedge B \subseteq A$.
Example: understanding equality
$A$ and $B$ are equal if they have the same elements.
$A=B \equiv \forall x. x\in A \leftrightarrow x\in B$
Which sets are equal to each other?
$A = \{1, 2, 3\}$
$B = \{3, 4, 5\}$
$C = \{3, 4\}$
$D = \{4, 3, 3\}$
$E = \{ 3, 4, 3\}$
$F = \{ 4, \{3\}\}$
$C, D, E$ are equal to each other.
Order of elements doesn’t matter, and duplicates don’t matter.
$F$ is not equal to $C,D,E$ because $\{3\}\neq 3$!
Example: understanding subsets
$A$ is a subset of $B$ if every element of $A$ is also in $B$.
$A \subseteq B \equiv \forall x. x\in A \rightarrow x\in B$
Example sets
$A = \{1, 2, 3\}$
$B = \{3, 4, 5\}$
$C = \{3, 4\}$
Are these subset formulas true or false?
$\varnothing\subseteq A$ $\mathsf{T}$
$A\subseteq B$ $\mathsf{F}$
$C\subseteq B$ $\mathsf{T}$
Building sets from predicates
$S = \{ x : P(x)\}$
$S$ is the set of all $x$ in the domain of $P$ for which $P(x)$ is true.
The domain of $P$ is often called the universe $U$.
$S = \{ x \in A : P(x)\}$
$S$ is the set of all $x$ in $A$ for which $P(x)$ is true.
Set operations
Set operations and their relation to Boolean algebra.
Union, intersection, and set difference
Union
$A\cup B = \{x : (x\in A)\vee(x\in B)\}$
Intersection
$A\cap B = \{x : (x\in A)\wedge(x\in B)\}$
Set difference
$A\setminus B = \{x : (x\in A)\wedge(x \not\in B)\}$
Given the following sets …
$A=\{1, 2, 3\}$
$B=\{3, 5, 6\}$
$C=\{3, 4\}$
Use set operations to make:
$[6]=$ $A\cup B\cup C$
$\{3\}=$ $A\cap B = A\cap C$
$\{1,2\}=$ $A\setminus B = A\setminus C$
Symmetric difference and complement
Symmetric difference
$A\oplus B = \{x : (x\in A)\oplus(x\in B)\}$
Complement (with respect to universe $U$)
$\overline{A} = \{x : x\not\in A\} = \{x:\neg(x\in A)\}$
Given the sets and universe …
$A=\{1, 2, 3\}$
$B=\{1, 2, 4, 6\}$
$U=\{1, 2, 3, 4, 5, 6\}$
What is
$A\oplus B=$ $\{3, 4, 6\}$
$\overline{A}=$ $\{4, 5, 6\}$
This is Boolean algebra again
Union $\cup$ is defined using $\vee$.
$A\cup B = \{x : (x\in A)\vee(x\in B)\}$
Intersection $\cap$ is defined using $\wedge$.
$A\cap B = \{x : (x\in A)\wedge(x\in B)\}$
Complement works like $\neg$.
$\overline{A} = \{x : x\not\in A\} = \{x:\neg(x\in A)\}$
This means that all equivalences from Boolean algebra translate
directly into set theory, and you can use them in your proofs!
DeMorgan’s laws
$\overline{A\cup B} = \overline{A}\cap\overline{B}$
$\overline{A\cap B} = \overline{A}\cup\overline{B}$
How would we prove these?
Proof technique:
To prove $C=D$, show
$x \in C \rightarrow x \in D$ and
$x \in D \rightarrow x \in C$.
Proof that $\overline{A\cup B} = \overline{A}\cap\overline{B}$:
Let $x\in \overline{A\cup B}$ be arbitrary.
Then, by definition of complement, $\neg (x\in A\cup B)$.
By definition of “$\cup$”, $\neg (x\in A \vee x\in B)$.
Applying DeMorgan’s laws, we get $x\not\in A \wedge x\not\in B$.
So, $x\in \overline{A} \wedge x\in \overline{B}$ by definition of complement.
Finally, $x\in \overline{A} \cap \overline{B}$ by definition of “$\cap$”, and
we have shown that $x\in\overline{A\cup B} \rightarrow x\in\overline{A}\cap\overline{B}$.
Next, let $x\in\overline{A}\cap\overline{B}$ be arbitrary.
$\ldots$
Finally, $x\in \overline{A\cup B}$, so $x\in\overline{A}\cap\overline{B} \rightarrow x\in\overline{A\cup B}$, which completes the proof. $\square$
Distributivity laws
$A \cap (B\cup C) = (A \cap B)\cup(A\cap C)$
$A \cup (B\cap C) = (A \cup B)\cap(A\cup C)$
A simple set proof
Prove that for any sets $A$ and $B$, we have $(A\cap B)\subseteq A$.
Recall that $X\subseteq Y \equiv \forall x. x\in X\rightarrow x\in Y$
Proof:
Let $A$ and $B$ be arbitrary sets and $x$ an arbitrary element of $A\cap B$.
Then, by definition of $A\cap B$, we have that $x\in A$ and $x\in B$.
It follows that $x\in A$, as required. $\square$
Set proofs can use Boolean algebra equivalences
Prove that for any sets $A$ and $B$, we have $(A\cap B) \cup (A\cap \overline{B}) = A$.
Proof:
Let $A$ and $B$ be arbitrary sets.
Since set operations are defined using logical connectives, the equivalences of Boolean algebra can be used directly, as follows:
$(A \cap B) \cup (A \cap \overline{B})$
$=$
$A \cap (B \cup \overline{B}) $
Distributivity
$=$
$A \cap U $
Complementarity
$=$
$A $
Identity
Universe $U$ corresponds to 1 and $\varnothing$ corresponds to 0.
More sets
Power set, Cartesian product, and Russell’s paradox.
Power set
Power set of a set $A$ is the set of all subsets of $A$.
$\mathcal{P}(A) = \{B : B\subseteq A\}$
Examples
Let $\text{Days}=\{M, W, F\}$.
$\mathcal{P}(\text{Days}) = $ $\{\varnothing,\{M\},\{W\},\{F\},\{M,W\},\{M,F\},\{W,F\},\{M,W,F\}\}$
$\mathcal{P}(\varnothing) = $ $\{\varnothing\} \neq \varnothing$
Cartesian product
The Cartesian product of two sets is the set of all of their ordered pairs.
$A\times B = \{(a,b): a\in A \wedge b \in B\}$
Examples
$\mathbb{R}\times\mathbb{R}$ is the real plane.
$\mathbb{Z}\times\mathbb{Z}$ is the set of all pairs of integers.
If $A=\{1, 2\}, B = \{a, b, c\}$,
then $A\times B = $ $\{(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)\}$.
$A\times\varnothing = $ $\{(a,b): a\in A \wedge b \in \varnothing\} = \{(a,b): a\in A \wedge \mathsf{F}\}=\varnothing$.
Russell’s paradox
Let $S$ be the set of all sets that don’t contain themselves.
$S = \{x: x\not\in x\}$
The definition of $S$ is contradictory, hence the paradox.
Suppose that $S \in S$. Then, by definition of $S$, $S\not\in S$, which is a contradiction.
Suppose that $S \not\in S$. Then, by definition of $S$, $S\in S$, which is a contradiction too.
To avoid the paradox …
Define $S$ with respect to a universe of discourse.
$S = \{x \in U: x\not\in x\}$
With this definition, $S\not\in S$ and there is no contradiction because $S\not\in U$.
Working with sets
Representing sets as bitvectors and applications of bitvectors.
Representing sets as bitvectors
Suppose that universe $U$ is $\{1,2,\ldots,n\}$.
We can represent every set $B\subseteq U$ as a vector of bits:
This is called the characteristic vector of set $B$.
Given characteristic vectors for $A$ and $B$, what is the vector for
$A \cup B = $ $(a_1\vee b_1) \ldots (a_n\vee b_n)$
$A \cap B = $ $(a_1\wedge b_1) \ldots (a_n\wedge b_n)$
Unix/Linux file permissions
$ ls -l
drwxr-xr-x ... Documents/
-rw-r--r-- ... file1
Permissions maintained as bitvectors.
Letter means the bit is 1.
”-“ means the bit is zero.
Bitwise operations
$\bitop{01101101}{\vee}{00110111}{01111111}$
z = x | y
$\bitop{00101010}{\wedge}{00001111}{00001010}$
z = x & y
$\bitop{01101101}{\oplus}{00110111}{01011010}$
z = x ^ y
Note that $(x\oplus y)\oplus y = x$.
Private key cryptography
Alice wants to communicate a message $m$ secretly to Bob, so that eavesdropper Eve who sees their conversation can’t understand $m$.
Alice and Bob can get together ahead of time and privately share a secret key $K$.
How can they communicate securely in this setting?
One-time pad
Alice and Bob privately share a random $n$-bitvector $K$.
Eve doesn’t know $K$.
Later, Alice has $n$-bit message $m$ to send to Bob.
Alice computes $C = m \oplus K$.
Alice sends $C$ to Bob.
Bob computes $m = C\oplus K$, which is $ (m \oplus K)\oplus K = m$.
Eve can’t figure out $m$ from $C$ unless she can guess $K$.
And that’s very unlikely for large $n$ …
Summary
Sets are a basic notion in mathematics and computer science.
Collections of objects called elements.
Can be compared for equality ($=$) and containment ($\subseteq$).
Set operations correspond to Boolean algebra operations.
You can prove theorems about sets using Boolean algebra laws.
Sets can be represented efficiently using bitvectors.
This representation is used heavily in the real world.
With this representation, set operations reduce to fast bitwise operations.