Lecture 5

We’ve seen in the above, that ABA\Rightarrow B and BAB\Rightarrow A are different statements. It’s helpful to have a word relating them:
Definition: Consider a statement of the form ABA\Rightarrow B. Then the converse of that statement is the statement BAB\Rightarrow A.

The truth of an implication has very little to do with the truth of its converse, as we’ll see.

Equivalence

Equivalence is the relationship between two statements of being both true, or both false.
Definition: Let PP and QQ be statements. We write PQP\Leftrightarrow Q, pronounced “PP is equivalent to QQ” for the statement that PP is true if and only if QQ is true.

Sometimes people shorten “if and only if” to “iff”.

Negation

Negation is a type of opposite:
Definition: Let PP be a statement. The negation of PP, written ¬P\neg P and often pronounced “not PP”, is the statement “PP is false”.

We must be careful in thinking of negation as an opposite. For example, the negation of “Richard is happy” is “Richard is not happy”.

Most people would say that the “opposite” is “Richard is sad”. That’s not quite the same thing!

Similarly, the negation of “Alice is in front of Bob” is not “Alice is behind Bob”, but:

“Alice is not in front of Bob”.

Note that double negation doesn’t do anything: the statement ¬(¬P)\neg(\neg P) is equivalent to PP. Since statements are either true or false, if it’s not “not true”, it’s true.

The negation allows us to make sense of something very important about implication:
Definition: Consider a statement of the form PQP\Rightarrow Q. Its contrapositive is the statement (¬Q)(¬P)(\neg Q)\Rightarrow(\neg P).

The main useful fact about the contrapositive is that it’s equivalent to the original implication. This is something very familiar from everyday life. If my friend Mel says

“If I can come visit you this evening, then I’ll text you at lunchtime,”

then I might rephrase it to myself as:

“I don’t get a text at lunchtime, then Mel won’t visit this evening.”

But here’s a formal statement and proof, anyway.

Proposition

Let PP and QQ be statements. The statement PQP\Rightarrow Q is equivalent to its contrapositive (¬Q)(¬P)(\neg Q)\Rightarrow(\neg P).

Proof

I’ll prove this using a truth table, showing what happens in all possibilities:

P  P\;   Q  \; Q\;   PQ  \; P\Rightarrow Q\;   ¬Q  \;\neg Q\;   ¬P  \;\neg P\;   (¬Q)(¬P)\;(\neg Q)\Rightarrow (\neg P)
0 0 1 1 1 1
0 1 1 0 1 1
1 0 0 1 0 0
1 1 1 0 0 1

You see from this that (¬Q)(¬P)(\neg Q)\Rightarrow(\neg P) is true exactly when PQP\Rightarrow Q is, and this proves that they’re equivalent.

“And” and “Or”

There are other ways to combine statements, other than implication:
Definition: Let PP and QQ be statements. The statement PQP\wedge Q, pronounced “PP and QQ”, is the statement that both PP and QQ are true.

The statement PQP\vee Q, pronounced “PP or QQ”, is the statement that at least one of PP or QQ (and possibly both) is true.

We can make truth tables for both of them:

PP QQ PQP\wedge Q
0 0 0
0 1 0
1 0 0
1 1 1
PP QQ PQP\vee Q
0 0 0
0 1 1
1 0 1
1 1 1

While you’re all used to these words from everyday life, there can be vagueness about how “or” is used in English.

For example,

you can have your pie with chips or with mashed potatoes

is probably intended to mean “but not both”. In mathematical argument when we use “or” and mean “but not both”, we have to say so explicitly.

It is sometimes worth knowing that implication can be defined in terms of “or”:

Proposition

The statement PQP\Rightarrow Q is equivalent to (¬P)Q(\neg P)\vee Q.

Proof

The only way that the first statement can be false is if PP is true and QQ is false. But that’s also the only way that the second statement can be false, so they’re equivalent.

Note that that shows another style of proof of logical statements: by analysis rather than the “case bash” used in truth tables.

Quantifiers

Lastly, we need to discuss how to make statements about general situations and particular examples. The phrases we use again and again are “for all” and “there exists”: these are called quantifiers.

The following symbols are in common use: for “for all”;for “there exists”;s.t.for “such that”.\begin{aligned} {}\forall \quad&\text{for ``for all'';}\\ {}\exists \quad&\text{for ``there exists'';}\\ {}\quad\mathrm{s.t.}\quad\quad&\text{for ``such that''.}\end{aligned}

So, for example, nZ,n21=(n+1)(n1)\forall n\in \mathbb{Z},\quad n^2-1 = (n+1)(n-1) is to be read as

“For all integers nn, we have n21=(n+1)(n1)n^2-1 = (n+1)(n-1).”

And xRs.t.x23x12=0\exists x\in\mathbb{R}\quad\mathrm{s.t.}\quad x^2-3x-12=0 is to be read as

“There exists a real number xx such that x23x12=0x^2-3x-12=0.”

It’s important that you get used to this notation. This is not because there is anything amazing about it, but because mathematics involves lots of general rules and particular examples: much of the mathematics you do for the next few years will require you to be able to deal with these things.

One thing you’ll have to get used to is situations with two or three quantifiers. These happen very frequently: “in general, there is always a particular example of such-and-such”, or “there is a particular amazing example which has the general property of such-and-such”.

For example, the statement nN,xRs.t.x2=n\forall n\in\mathbb{N},\quad\exists x\in\mathbb{R}\quad\mathrm{s.t.}\quad x^2 = n says that every natural number nn has a square root xx.

The order of quantifiers is very important. If we swap over the two quantifiers in the last example, we get xRs.t.nN,x2=n.\exists x\in\mathbb{R}\quad\mathrm{s.t.}\quad\forall n\in\mathbb{N},\quad x^2 = n. This says that there’s a particular number xx which has the property that xx is the square root of every natural number. And that’s nonsense.

Another thing that mathematicians have to do every day is understanding how negation interacts with quantifiers.

The negation of “all Teletubbies are red”is “not all Teletubbies are red”, which is equivalent to “there exists a Teletubby which is not red”.

Similarly, the negation of “there exists a dolphin who likes Beethoven”is “there does not exist a dolphin who likes Beethoven”,and that’s equivalent to “all dolphins do not like Beethoven”.

In symbols, ¬(xX,P(x))is equivalent toxXs.t.¬P(x).¬(xXs.t.P(x))is equivalent toxX,¬P(x).\begin{aligned} {}\neg\left(\forall x\in X,\quad P(x)\right) \qquad&\text{is equivalent to}\qquad \exists x\in X\quad\mathrm{s.t.}\quad\neg P(x).\\ {}\neg\left(\exists x\in X\quad\mathrm{s.t.}\quad P(x)\right) \qquad&\text{is equivalent to}\qquad\forall x\in X,\quad \neg P(x).\end{aligned}

Perhaps you may want to remember that “negation swaps \forall and \exists.” But being able to do it correctly by remembering what’s going on is much more important than remembering a slogan. After a while it should come to seem natural.

Suppose I am wondering whether all fish are slippery. If it’s true, I need to find some general reason why every single fish is slippery. If it’s false, I only need to find one single fish which isn’t slippery, and then I’ve proved it.

In general, if you have a general statement and you don’t know if whether it’s true or false, then it could either be: