Nested Quantifiers

Lecture 9, DSCI 220, 2026W1

Announcements

• EX1 is marked. EX2 is Thu 10/8 through Sun 10/11.
• Practice problems for EX2 go up Wednesday night.
• No class Wednesday — National Day for Truth and Reconciliation.
• HW2 is out, due Sun Oct 4.

Warm-up: Introduce yourself to a classmate you haven’t met yet. Ask them one question about their life, and tell them one thing about yours.

Code Patterns

The loops ARE the definitions. forall walks the domain looking for a counterexample and returns the first it finds; exists walks it looking for a witness. Blanks:

forall: if not P(x): return False, x then return True, None exists: if P(x): return True, x then return False, None

Note what each returns on failure – a counterexample for forall, nothing for exists. That asymmetry is the Counterexample and Witness slides of lecture 8, in code.

Nested Quantifiers

More Than One

Friday we bound one variable. Nothing stops us binding two.

     \(\forall x\, \forall y\ \text{Knows}(x,y)\)

     \(\forall x\, \exists y\ \text{Knows}(x,y)\)

     \(\exists x\, \forall y\ \text{Knows}(x,y)\)

Say each one out loud. ______________________________

Everybody knows everybody. / Everybody knows somebody. / Somebody knows everybody.

The third is the one worth slowing on – it is a claim about ONE person, and it is much stronger than the second. That is the whole lecture in one slide, so do not resolve it yet; just let them hear that two and three are different.

Every Lock Has a Key

\(K(k, l)\): key \(k\) opens lock \(l\).

	Says what, in English?
\(\forall l\, \exists k\ K(k,l)\)
\(\exists k\, \forall l\ K(k,l)\)

Which one would a locksmith charge you more for? ____________

Top: every lock has some key that opens it. Ordinary. Bottom: there is one key that opens every lock. A MASTER KEY.

Same two symbols, same relation, swapped order – and one of them is worth paying for. Nobody needs convincing that these differ once it is a master key.

Then the direction that does hold: a master key IS a key for each lock, so the bottom gives you the top. The top gives you nothing about the bottom.

Does Order Matter?

Swap	Same?
\(\forall x\, \forall y\ P(x,y)\)   vs   \(\forall y\, \forall x\ P(x,y)\)
\(\exists x\, \exists y\ P(x,y)\)   vs   \(\exists y\, \exists x\ P(x,y)\)
\(\forall x\, \exists y\ P(x,y)\)   vs   \(\exists y\, \forall x\ P(x,y)\)

So the rule is: __________________________________

Yes, yes, NO.

Two quantifiers of the SAME kind commute freely. Two of DIFFERENT kinds do not. Verified exhaustively over every relation on a three-element domain.

And the asymmetry is one-way:

exists y forall x  ==>  forall x exists y        always
forall x exists y  ==>  exists y forall x        NOT

Of the 512 relations on a domain of three, 174 satisfy the second and fail the first. It is the common case, not a curiosity.

One Picture

Rows are \(x\), columns are \(y\). A mark means \(P(x,y)\) holds.

	\(y_1\)	\(y_2\)	\(y_3\)
\(x_1\)			✓
\(x_2\)			✓
\(x_3\)		✓

Every row has a mark, so _____________________

No column is all marks, so _____________________

forall x exists y is TRUE – every row has something. exists y forall x is FALSE – no single column is full.

This is the picture to keep. “Everyone has one” is a statement about ROWS. “One works for everyone” is a statement about COLUMNS. Swapping the quantifiers swaps which way you read the grid, and a grid can easily have every row covered with no column full.

Draw the master key on it: a master key is a FULL COLUMN.

Your Turn

In pairs, work the worksheet.

Page 3 is the grid you just saw — except now you place the hearts.

activities/lecture-09_activity.pdf, four pages, linked from the schedule.

1 Eight students, seven predicates. For five statements: English, truth value, and the NEGATION. The negations are the part that will slow them. 2 They write their own – one true, one false, two predicates minimum, plus negations. 3 Hero/Villain grids. Place hearts to realise a statement using AS FEW AS POSSIBLE, then judge statements against a given grid. 4 A new quantifier meaning “there exists exactly 2”. Minimum and maximum hearts to make each statement true.

Page 3 is One Picture made constructive: reading a grid is easy, BUILDING the sparsest one that satisfies forall-exists forces them to notice it needs one mark per row, while exists-forall forces a whole column. Point back to the slide when you set them off.

Page 4 is the good one, and it is the reason not to rush. “Exactly 2” is a quantifier they have never met, and min/max makes them reason about the SHAPE of a satisfying grid rather than recall a rule.

TIMING. Four pages is more than the slot holds. Roughly 25 minutes gets them through 1 and 3. If you want page 4 – and it is the most interesting – cut page 2, which is the most easily set as homework.

Words and Ideas

Nested quantifiers

Order of quantifiers

\(\forall\exists\) versus \(\exists\forall\)