Mathematical Statements

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10 illustrated lessons, each teaching the why before the how.

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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

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what it isdefinitionfixes a meaningpropositiona claim, true or falsetheorema claim with a proof

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

n is even means n = 2k for some whole k

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

every even number above 2 is a sum of two primes

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

in a right triangle,
a proof existsand here is an argument that it holds

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

Now you

a prime has exactly two whole factors

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

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PQ1TT2TF3FT4FF

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

PQP and Q1TTT2TFF3FTF4FFF

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

PQP and QP or Q1TTTT2TFFT3FTFT4FFFF

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

P: n is eventhe truth value flipsnot P: n is odd

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

Now you

PQcaseTT

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

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P: x > 5deny P, and nothing further
not P:

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

P: x > 5this denies too muchx < 5

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

not (P and Q)at least one of them fails(not P) or (not Q)

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

not (P or Q)neither one holds(not P) and (not Q)

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

Now you

the shape is a square and blue

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

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if n is a multiple of 4, then n is eventhe part assumedhypothesis: n is a multiple of 4the part promisedconclusion: n is even

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

PQif P then Q1TTT2TFF3FTT4FFT

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

n = 6: not a multiple of 4the hypothesis failedthe promise says nothing about 6

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

n = 6 is even, but 6 is not a multiple of 4a conditional works one way onlyso 6 is not a counterexample

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

Now you

if a shape is a square, it has four sides

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

if it rained, the ground is wet

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

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saysoriginalif P then Qconverseif Q then P

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

trueif n is a multiple of 4, then n is evenfalse at n = 6if n is even, then n is a multiple of 4

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

saysoriginalif P then Qconverseif Q then Pinverseif not P then not Q

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

if n is not a multiple of 4, then n is not eventhe same counterexample worksfalse at n = 6 as well

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

Now you

if a shape is a square, then it has four sides

What is the inverse of the statement drawn?

if a shape is a square, then it has four sides

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

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saysoriginalif P then Qconverseif Q then Pinverseif not P then not Qcontraposif not Q then not P

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

PQP→Q¬Q→¬P1TTTT2TFFF3FTTT4FFTT

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

trueif n is a multiple of 4, then n is eventrue, and for the same reasonif n is odd, then n is not a multiple of 4

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

sayspair oneoriginal, contrapositivepair twoconverse, inverse

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

Now you

saysconverseif Q then Pinverseif not P then not Qcontraposif not Q then not P

Which statement always agrees with the converse?

saysconverseif Q then Pinverseif not P then not Qcontraposif not Q then not P

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

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if n is a multiple of 4, then n is evensufficientbeing a multiple of 4 is enough

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

if n is a multiple of 4, then n is evennecessarybeing even cannot be avoided

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

meanssufficientenough on its ownnecessarycannot be done without

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

even is necessary for a multiple of 4so it is not sufficientbut 6 is even and not a multiple of 4

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

Now you

a triangle has three equal sides exactly when it has three equal angles

What are three equal angles, for three equal sides?

if n is a multiple of 4, then n is even

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

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P if and only if Qtwo claims in one sentenceif P then Q, and if Q then P

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

saysforwardif P then Qbackwardif Q then P

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

n is even if and only if is even
the easy direction
n even even
the direction that needs work
even even

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

n is a multiple of 4 if and only if n is eventhe backward direction failsfalse: 6 is even

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

Now you

a whole number is odd if and only if it leaves remainder 1 on division by 2

What is the status of the statement drawn?

n is a multiple of 4 if and only if n is even

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

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for all n,
the universal quantifierevery single n, with no exception

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

there exists n with
the existential quantifierone such n is enough

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

proved bybroken byfor allan argumentone exampleexistsone examplean argument

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

for all n, there exists m with m > nthe order was changedthere exists m, for all n, m > n

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

Now you

for all whole n, n is even

What would break the statement drawn?

for all whole n,

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

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for all n, P(n)the quantifier swaps, the inside is deniedthere exists n with not P(n)

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

there exists n with P(n)the swap runs the other way toofor all n, not P(n)

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

not: all primes are oddand 2 is that primesome prime is not odd

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

not: all primes are oddthis denies far too muchno prime is odd

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

Now you

there exists a whole n with

What is the negation of the statement drawn?

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