www.carom-maths.co.uk. 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.
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Activity 2-5: What are you implying?
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.
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
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?
1 is (I, I),
3 and 5 are (RO, RO),
2 and 4 are (NINRO, NINRO),
while6 is (I, NINRO).
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.
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?
if you had to assign
and (I, NINRO)
you do it?
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, firstname.lastname@example.org