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.
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