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Activity 2-5: What are you implying?

You are given these four cards:

Each of these cards has a letter on one side

and a digit on the other.

This rule may or may not be true:

if a card has a vowel on one side,

then it has an even number on the other.

Which cards must you turn

to check the rule for these cards?

that has tested people’s grasp of logic for years.

Most people pick Card with A, correctly.

Many pick Card with 2 in addition.

But… if the other side of the Card with 2 is a vowel, that’s fine.

And if the other side of the Card with 2 is a consonant,

that’s fine too!

The right answer is to pick Card with 7 in addition to the A.

If the other side of the 7 is a vowel, there ISa problem.

Consider these four assertions:

In the statements below, a, n and m are positive integers

1. a is even

2. a2 is even

3. a can be written as 3n + 1

4. a can be written as 6m + 1

If we make a card with one statement on the front and one on the back, there are six possible cards we could make.

Define A I B to signify ‘A implies B’,

Define A RO B to signify ‘A rules out B’, and

A NINRO B to mean ‘A neither implies nor rules out B’.

We can see that the statements for Card 1

mean that this is of type (I, I): each side implies the other.

Task: how many different

types of card do we have

with these definitions?

We conclude we have four different types of card:

1 is (I, I),

3 and 5 are (RO, RO),

2 and 4 are (NINRO, NINRO),

while6 is (I, NINRO).

Are these the only possible types of card?

Yes, since if A RO B, then B RO A,

so (RO,I) and RO, NINRO)

are impossible cards to produce.

Can we be sure that if A RO B, then B RO A?

We can use a logical tautology called MODUS TOLLENS:

if A implies B, then (not B) implies (not A).

Now A RO B means A I (not B).

If A I (not B), then by Modus Tollens,

not (not B) implies (not A).

Thus B implies (not A), that is B RO A.

if A implies B, then (not B) implies (not A).

What mistake do people make with the four-card problem?

They assume that A I B can be reversed to B I A.

In fact, A I B CAN be reversed,

but to (not B) I (not A) – Modus Tollens.

So VowelIEven reverses to (Not Even) I (Not Vowel),

which shows we need to pick the 7 card.

Let’s try a variation on our initial four-card problem.

This time the obvious reversal works, for ‘Tree RO Panda’ is the same as ‘Panda RO Tree’: you DO need to turn over the two cards named in the question.

You are given four cards below.

Each has a plant on one side and an animal on the other.

Task: you are given the rule:

if one side shows a tree, the other side is not a panda.

Which cards do you need to turn over to check the rule?

Given two circles, there are four ways that they can lie in relation to each other.

Task:

if you had to assign

(I, I),

(RO, RO),

(NINRO, NINRO),

and (I, NINRO)

to these,

how would

you do it?

Let’s invent a new word, DONRO,

standing for DOes Not Rule Out.

Task: is it true that

if A DONRO B and B DONRO C, then A DONRO C?

Pick an example to illustrate your answer.

A: The shape S is a red quadrilateral

B: The shape S is a rectangle

C: The shape S is a blue quadrilateral

A: x = 2

B: x2 = 4

C: x = 2

A: ab is even

B: b is less than a

C: ab is odd

The Open University and my teachers on the

Researching Mathematical Learning course.

Mathematics In School

for publishing my original article on this subject.

Carom is written by Jonny Griffiths, [email protected]

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