In this section, we’ll introduce the basics of propositional logic. Even if you’re unfamiliar with the idea, you’ve almost certainly used it in programming before.

`\E`

`\E`

means “exists”. We write `\E x \in S : P(x)`

to say “there is at least one x in the set such that P(x) is true.” It’s the equivalent of a python `any()`

or a Ruby `any?`

. If we wanted to check that a set had an even number in it, we could write `HasEvenNumber(S) == \E x \in S : IsEven(x)`

`~\E`

is the opposite: it says that there are no such elements. `\E x \in {}: Foo`

is always false, since there are no elements in `{}`

.

Assuming `Sum(S)`

returns the sum of the elements of S, write an operator that, for a given set of integers S and integer N, returns if there are N elements in S that sum to 0. eg

```
SumsToZero({1, 2, -7, 4, 11}, 2) = FALSE
SumsToZero({1, 2, -7, 4, 11}, 4) = TRUE
SumsToZero({1, 2, -7, 4, 11}, 5) = FALSE
```

```
SumsToZero(S, N) == \E s \in SUBSET S:
/\ Cardinality(s) = N
/\ Sum(s) = 0
```

`\A`

`\A`

means “all”. We write `\A x \in S : P(x)`

to say “For every x in the set, P(x) is true.” If we wanted to check that a set had no odd numbers in it, we could write `OnlyEvenNumbers(S) == \A x \in S : IsEven(x)`

. If there are only even numbers, `HasEvenNumber`

is true. Otherwise it’s false. Simple.

`~\A`

is the opposite: it says that there is at least one element where P(x) is false. `\A x \in {}: Foo`

is always true, since there are no elements in `{}`

, so all zero elements pass the test.

`\A x \in {}: FALSE`

is still true!

Given a set and an operator, determine whether the operator is commutative over *all* elements in the set.

`IsCommutative(Op(`*,*), S) == \A x \in S :
\A y \in S : Op(x,y) = Op(y,x)

Alternatively, we could put them on the same line:

`IsCommutative(Op(`*,*), S) == \A x \in S, y \in S : Op(x,y) = Op(y,x)`

`=>`

and `<=>`

`P => Q`

means “If P is true, then Q must also be true.” Note that P can be false and Q can be true, or both can be false. It’s equivalent to writing `~P \/ Q`

, which is how TLC interprets it.

`P <=> Q`

means “Either both P and Q are true or both are false.” It’s equivalent to writing `(~P /\ ~Q) \/ (P /\ Q)`

. `P <=> ~Q`

is P XOR Q.

Without looking back at the introduction, write an operator that returns the maximum number of a set.

```
Max(S) == CHOOSE x \in S : \A y \in S : y <= x
```

Both `=>`

and `<=>`

follow the same precedence rules as logical junctions. In other words, TLC interprets

```
/\ A
/\ B
=> C
```

as `A /\ (B => C)`

, whereas without the indent it’s interpreted as `(A /\ B) => C`

.

While we introduced the CHOOSE operator back in sets, it really comes into its own when we add the logical operators. Many quantified properties, such as “the largest x such that P”, can be expressed as “the x where all larger elements don’t have P” or “the x where all of the other elements with P are smaller”. For example, what is the largest prime in a set S?

```
IsPrime(x) == x > 1 /\ ~\E d \in 2..(x-1) : x % d = 0
LargestPrime(S) == CHOOSE x \in S:
/\ IsPrime(x)
/\ \A y \in S:
IsPrime(y) => y <= x
\* or y > x => ~IsPrime(y)
```

A prime number p is a *twin prime* if p-2 is prime or p+2 is prime. Find the largest twin prime in S.

```
IsTwinPrime(x) == /\ IsPrime(x)
/\ \/ IsPrime(x + 2)
\/ IsPrime(x - 2)
```

```
LargestTwinPrime(S) == CHOOSE x \in S:
/\ IsTwinPrime(x)
/\ \A y \in S: IsTwinPrime(y) => y <= x
* or y > x => ~ IsTwinPrime(y)
```

Now return the largest pair of twin primes, ordered by value. Assume that S may be missing numbers and, if one of the twin primes is missing, the pair is invalid. For example, the largest pair in `{3, 5, 13}`

is `<<3, 5>>`

, not `<<5, 13>>`

.

```
LargestTwinPair(S) == CHOOSE <<x, y>> \in S \X S:
/\ IsPrime(x)
/\ IsPrime(y)
/\ x = y - 2
/\ \A <<w, z>> \in S \X S:
/\ IsPrime(z)
/\ IsPrime(w)
/\ w = z - 2
=> z < y
```

Given `stockprices`

is a tuple of positive integers representing the value of a stock at a given time of day, write an operator that determines the maximum profit you could make by buying and selling a single stock. Assume for this problem that you cannot short; you must buy before you sell.

```
MaxProfit(stockprices) ==
LET sp == stockprices \* clean it up a bit
TimePair == (1..Len(sp)) \X (1..Len(sp))
Profit[p \in TimePair] == sp[p[2]] - sp[p[1]]
best == CHOOSE best \in TimePair :
/\ best[2] > best[1] \* Buy after sell
/\ Profit[best] > 0 \* Make money plz
/\ \A worse \in TimePair :
worse[2] > worse[1] => Profit[best] >= Profit[worse]
IN Profit[best]
```

Note this will crash if there is no possible pair, which is preferrable to paying trading fees twice on a zero-dollar profit.