Speak Maths · Last school years

Logic: every door has a key

Mathematics is careful about 'and', 'or', 'if', 'all' and 'some'. A few signs make each one exact, so a sentence can be true or false and nothing in between.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$\neg P$ Say: not PMeans: turns true into false and false into true
$P \land Q$ Say: P and QMeans: true only when both are true
$P \lor Q$ Say: P or QMeans: true when at least one is true; both is allowed
$P \Rightarrow Q$ Say: P implies Q; if P then QMeans: whenever P is true, Q is true too; if P is false it says nothing
$P \Leftrightarrow Q$ Say: P if and only if QMeans: P and Q are both true or both false
$\forall x P (x)$ Say: for all x, P of xMeans: P of x is true for every x
$\exists x P (x) \exists! x P (x)$ Say: there exists an x with P of x; there exists exactly one x with P of xMeans: at least one x; the exclamation mark makes it exactly one
$∴ Q$ Say: therefore QMeans: Q follows from what was said before

The story: Every door has a key

Subtitles:

Tap a line to see its words.

$\forall d K (d)$ A notice says: every door has a key.
$K (1) \land K (2) \land K (3)$ Doors 1, 2 and 3 each have a key.
$\neg K (4)$ Door 4 has no key.
$∴ \neg\forall d K (d)$ So the notice is not true.
$\exists d \neg K (d)$ There is a door with no key.
$\exists! d \neg K (d)$ There is exactly one such door: door 4.
$L (1) \land L (2) \land \neg L (3) \land \neg L (4)$ Doors 1 and 2 are locked; doors 3 and 4 stand open.
$L (1) \lor K (4)$ Door 1 is locked or door 4 has a key. The first part is true, so the sentence is true.
$\neg (L (3) \Leftrightarrow K (3))$ Door 3 has a key but is not locked, so 'locked' and 'has a key' do not always go together.
$\forall d (L (d) \Rightarrow K (d))$ A new notice: every locked door has a key. True! Door 4 has no key, but it is not locked.

In words this story is 505 letters long; in symbols, 102.

Read it

What does it say?

$\neg\forall d K (d)$

$P \Rightarrow Q$

$\exists! x x + 2 = 5$

$\neg (P \land Q)$

$\neg P \lor Q$

$\forall x \exists y y > x$

$\exists y \forall x y > x$

$P \Leftrightarrow \neg Q$

$\neg\exists x P (x)$

$(P \Rightarrow Q) \land \neg Q$

$\exists! x (x > 0 \land x^{2} = 4)$

$\neg (P \lor Q)$

Write it

Which notation says this?

Some door has no key.

For every number there is a bigger number.

P is true exactly when Q is true.

If it rains, the street is wet. (R: it rains; W: the street is wet.)

There is exactly one x with x plus 1 equal to 3.

No door is open. (O(d): door d is open.)

P is false and Q is false.

There is one key that opens every door. (O(k, d): key k opens door d.)

Every door is opened by some key, perhaps a different one each time. (O(k, d): key k opens door d.)

P or Q is true, and P is false; therefore Q.

A number is even if and only if it is not odd. (E(n): n is even; O(n): n is odd.)

Not every x has property B.

Build it

Tap the tiles in order. One tile does not belong.

P and Q, therefore P.

For every x there is exactly one y with y equal to two x.

If P then Q; P; therefore Q.

There is no x with P of x.

For every x, if x is even then x squared is even. (E: is even.)

P or Q, and not both.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 The corridor of doors

Three doors, three keys. Judge each sentence in this corridor: true or false?

	$k_{1}$	$k_{2}$	$k_{3}$
$d_{1}$	✓
$d_{2}$	✓	✓
$d_{3}$			✓

A tick means O(k, d): key k opens door d.

$\forall d \exists k : O (k, d)$
$\exists k \forall d : O (k, d)$
$\forall k \exists d : O (k, d)$
$\exists! k : O (k, d_{2})$
$O (k_{1}, d_{3}) \lor O (k_{3}, d_{3})$
$\neg\exists k : O (k, d_{1}) \land O (k, d_{3})$

2 Who likes whom

L(x, y) means x likes y. Pair each line with what it says.

Tap a line, then its meaning.

3 Not every bird flies

This should say: “not every bird flies” is the same as “some bird does not fly”. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 The one and only

Exactly one positive whole number fits. Which?

$\exists! n \in N : n > 0 \land n + n = n \cdot n$

5 Say the opposite

Which sentence says the same as this negation, without the ¬ in front?

$\neg (\forall x \exists y : x < y)$

6 The detective

R: it rained. W: the street is wet. The detective knows both lines. What follows?

$R \Rightarrow W, \neg W ∴ ?$

7 Write the proverb

Every problem has a solution, but no single solution fits every problem. Write it in symbols, with S(s, p) for “s solves p”. Then compare.

Your version

8 Your own theorem

A whole number is even exactly when its square is even. Write it in symbols, using 2 ∣ n for “2 divides n”.

Your version

9 Switches and lamps

Three switches, three lamps. Judge each sentence in this room: true or false?

	$l_{1}$	$l_{2}$	$l_{3}$
$s_{1}$	✓	✓
$s_{2}$		✓
$s_{3}$		✓	✓

A tick means C(s, l): switch s controls lamp l.

$\exists l \forall s : C (s, l)$
$\exists s \forall l : C (s, l)$
$\forall l \exists s : C (s, l)$
$\exists! s : C (s, l_{1})$
$C (s_{2}, l_{1}) \lor C (s_{2}, l_{3})$
$\forall s (C (s, l_{2}) \Rightarrow C (s, l_{1}))$

10 The night of the broken vase

Three guests, three rooms, one broken vase. The detective’s notes say who entered which room. Judge each line: true or false?

	$r_{1}$	$r_{2}$	$r_{3}$
$g_{1}$	✓
$g_{2}$	✓	✓
$g_{3}$			✓

A tick means I(g, r): guest g was in room r.

$\forall g (I (g, r_{2}) \Rightarrow I (g, r_{1}))$
$\forall g (I (g, r_{1}) \Leftrightarrow I (g, r_{2}))$
$\exists! g : I (g, r_{3})$
$\neg\exists g (I (g, r_{1}) \land I (g, r_{3}))$
$\exists g \forall r : I (g, r)$
$I (g_{3}, r_{1}) \lor I (g_{3}, r_{2})$

11 Saying no

Five sentences that begin with “not”. Pair each with what it really says.

Tap a line, then its meaning.

12 Which way does the arrow point?

Four arrows between P and Q. Pair each with its meaning.

Tap a line, then its meaning.

13 Exactly one

This should say: there is exactly one natural number n with n + 3 = 5. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 Neither

This should say: “not (P or Q)” is the same as “not P and not Q”. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

15 The rule and the case

Every x with P has Q, and a has P. What follows about a?

$\forall x (P (x) \Rightarrow Q (x)), P (a) ∴ ?$

16 One way only

Every number divisible by 4 is even, but not every even number is divisible by 4. Write it in symbols, using 2 ∣ n for “2 divides n”. Then compare.

Your version

Tell your own

Tell a three-line story about a rule that turns out false for one case, in signs only. Use 'for all' or 'there exists' once. Then read it aloud.

Kept on this device only.

Logic: every door has a key

The alphabet

The story: Every door has a key

Read it

Write it

Build it

Challenges

1 The corridor of doors

2 Who likes whom

3 Not every bird flies

4 The one and only

5 Say the opposite

6 The detective

7 Write the proverb

8 Your own theorem

9 Switches and lamps

10 The night of the broken vase

11 Saying no

12 Which way does the arrow point?

13 Exactly one

14 Neither

15 The rule and the case

16 One way only

Tell your own

Download for offline use