Mathematical Statements

Definitions, Propositions and Theorems #

A meaning fixed, a claim made, a claim proved.

A definition fixes a meaning, a proposition claims something, and a theorem is a proposition already proved

Mathematics uses three kinds of sentence, and they do three different jobs.

A definition only fixes a meaning. There is nothing in it to agree or disagree with.

A proposition claims something. This one has no proof and no counterexample yet.

A theorem is a proposition someone has proved. Until a proof exists, it is only a claim.

Now you

What kind of sentence is on the board?

Logical Connectives and Truth Tables #

Every case for and, or and not, in one table.

A truth table lists every case at once, which settles exactly what and, or and not do

Each statement is true or false. Two statements give four cases, and no more.

P and Q is true in one case only: the row where P and Q are both true.

P or Q is true in three cases. In mathematics or includes the case where both hold.

Not needs one statement, not two. It is true exactly when the statement is false.

Now you

Which of these is true in the case drawn?

Negating a Statement #

Deny exactly what was claimed, and no more.

The negation of a statement is true exactly when the statement is false, and it must deny no more than that

The negation of x > 5 is x ≤ 5, because equality was never ruled out.

Flipping the sign to x < 5 loses the case x = 5, so it denies more than P claimed.

Denying that both hold only says at least one fails. And becomes or under a negation.

Denying that either holds says neither holds, so or becomes and instead.

Now you

What is the negation of the statement drawn?

Conditional Statements #

One case breaks the promise, and only one.

A conditional promises that whenever its hypothesis holds its conclusion holds, and nothing at all otherwise

A conditional has two parts: the hypothesis you assume, and the conclusion it promises.

Only one case breaks the promise: the hypothesis is true and the conclusion is false.

When the hypothesis fails, the conditional makes no claim and cannot be broken.

An even number that is not a multiple of 4 does not test this claim at all.

Now you

Take the case: a rectangle that is not a square. What does that do to the statement drawn?

Take the case: it did not rain and the ground is wet. What does that do to the statement drawn?

The Converse and the Inverse #

Swap the halves, or negate them in place.

Swapping the two parts of a conditional gives its converse, and negating both gives its inverse, and those two always agree

The converse swaps the hypothesis and the conclusion. That makes a different claim.

Here the original is true and the converse is false. The counterexample is n = 6.

The inverse negates both parts instead of swapping them, and keeps them in order.

The same counterexample breaks both, because converse and inverse say the same thing.

Now you

What is the inverse of the statement drawn?

What is the converse of the statement drawn?

The Contrapositive #

Swap and negate, and the truth is unchanged.

Swapping the two parts of a conditional and negating both gives a statement that is true exactly when the original is

The contrapositive does both moves: swap the two parts, then negate each of them.

The two columns agree in all four cases, so the two statements say the same thing.

Here the contrapositive is true whenever the original is. The converse was not.

So the four statements fall into two pairs, and the pairs need not agree with each other.

Now you

Which statement always agrees with the converse?

Which of the three has to be true whenever the original is?

Necessary and Sufficient Conditions #

Enough on its own, or impossible to avoid.

In a true conditional the hypothesis is sufficient for the conclusion and the conclusion is necessary for the hypothesis

Being a multiple of 4 is enough to make n even. Nothing else has to be checked.

Being even is unavoidable for a multiple of 4. It is required, but it is not enough.

Sufficient means enough on its own. Necessary means you cannot do without it.

A condition can be necessary without being sufficient, and n = 6 shows it here.

Now you

What are three equal angles, for three equal sides?

What is being even, for being a multiple of 4?

If and Only If #

Two implications claimed in one sentence.

If and only if claims a conditional and its converse at once, so proving it means proving both directions

If and only if states two conditionals at once, one running in each direction.

So a proof has two parts, and leaving either one out leaves the claim unproved.

Here one direction is a line of algebra and the other needs the contrapositive.

One direction is never enough: 6 is even, but 6 is not a multiple of 4.

Now you

What is the status of the statement drawn?

Which single number settles the statement drawn?

For All and There Exists #

Every case, against merely some case.

For all claims something of every case while there exists claims only that one case can be found

For all claims something of every case at once, so one exception destroys it.

There exists claims only that some case can be found, so one example settles it.

A for all claim is proved by an argument and broken by one example. Exists is the reverse.

Order matters: the first statement is true of whole numbers, the second statement is false.

Now you

What would break the statement drawn?

What would settle the statement drawn?

Negating a Quantified Statement #

Not all means at least one fails.

Negating a quantified statement swaps the quantifier over and negates what it was claiming

Denying that every case works says only that at least one case fails.

Denying that any case works says every case fails, so exists becomes for all.

Not all primes are odd. The negation says some prime is not odd, and 2 is that prime.

Negating only the inside gives "no prime is odd", which claims far more than needed.

Now you

What is the negation of the statement drawn?