The Difference between And and Or

Date: 01/23/2008 at 11:59:54
From: Mikki
Subject: and/or

My son had a question that was marked wrong on his paper.  He pointed
out to me that by the way it was worded, he felt as though he were
correct.  Here is the question:  There are 3 knives, 4 spoons, 4
forks.  What fraction of the utensils are spoons OR forks?

He answered 4/11 and was told the teachers edition says 8/11.  I
understand the way he read it to be OR meaning one or the other.  If
it's 8/11, shouldn't it be worded spoons AND forks?  If the answer is
8/11, I want my son to understand why.

Date: 01/23/2008 at 12:24:17
From: Doctor Peterson
Subject: Re: and/or

Hi, Mikki.

The words "and" and "or" can be ambiguous in English, so in math we 
give them precise meanings.  We have to teach those meanings, but 
often forget to, which may have happened here.

When we talk about the set of things that are A AND B, we mean that 
EACH of those things must be BOTH A and B.  Nothing is both a spoon 
and a fork!  (At least not in this problem.)  So "and" would have been 
inappropriate.  There are no utensils that are spoons and forks.

When we talk about the set of things that are A OR B, we mean that 
EACH of them may be EITHER A or B.  That is, we are including in the 
set BOTH those that are A, AND those that are B.  This is where the 
confusion and ambiguity come in!  There are 8 utensils that are 
spoons or forks.

Your son read it in a way that is commonly used in nontechnical 
English, taking "How many are A or B" to mean two separate questions 
combined: "How many are A, how many are B".  I can see how that could 
be tempting in this case; the two numbers happen to be the same, so 
he could take the question to mean "How many are A (which is also 
the same as the number that are B".  If there had been 3 spoons and 4 
forks, that interpretation would not have made as much sense; the best 
answer he could give would be "3, or 4".  We don't combine questions 
like that in math, to avoid confusion.

So the book was right, but the question is ambiguous if the teacher 
has not taught (or does not know) the standard mathematical usage. 
(This usage is important in some later topics, such as probability, 
so it's definitely worth teaching.)

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/23/2008 at 12:47:17
From: Doctor Riz
Subject: Re: and/or

Hi Mikki -

I'd like to add one piece to what Dr. Peterson wrote.  While this is a
slightly different application of the idea, my students always found
this particular example of the logical difference between AND and OR
helpful.  

In logic, an AND statement is only true if both parts of it are true.
If I say, "I am in Vermont AND I am in New Hampshire" the only way
that can be true is if I am standing on the border with one foot in
each state.

An OR statement is true if either part is true.  If I say, "I am in
Vermont OR I am in New Hampshire" that statement is true as long as I
am in either state (it's also true if I'm straddling the border).  The
only way an OR statement is not true is if both parts are false, such
as if I were standing in Massachusetts when I made my statement about
being in Vermont or New Hampshire.

With your question about utensils being spoons OR forks, I count every
utensil that is either a spoon or a fork, giving 8 of the 11.  If I
were asked what fraction of the utensils were spoons AND forks, there
would be zero since the utensil would have to be both things.  There
IS a utensil you sometimes see in fast food places which is a spoon
shape with teeth on the front edge, and it's generally referred to as
a "spork", a combination of spoon and fork.  That's what I'd need for
a utensil to be considered a spoon AND a fork.

Does that help?  Write back if you have questions on any of this.

- Doctor Riz, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/23/2008 at 13:33:11
From: Mikki
Subject: Thank you (and/or)

Thank you both for your prompt responses.  You have taught me 
something and definitely given me something to think about.