Monday, March 3, 2008

Mathematical "OR"

Mathematicians tend to be humans (exceptions unknown). Because of this they are forced to use the same languages as those around them. Some words, however, have different meaning when used in mathematics. A perfect example of this is the word "or".

The following stories (they are real) show this difference very well:

Sandwich Shop Worker: “Would you like mustard or mayonnaise on your sandwich?”

Customer: “Yes, please.”

(Sandwich shop worker stares)

Customer: “Oops! I mean. Um…mayonnaise.”

Sandwich Shop Worker: “How would you like to pay? Cash or credit?”

Customer: “Okay.”

And a second one:

Customer: “Can I have the giant Yorkshire Pudding?”

Worker: “Sure, would you like it served with Beef or Pork?”

Customer: “Yes, I’ll have beef or pork.”

Worker: “No, would you like Beef OR Pork?”

Customer: “YES, beef or pork!”

It may sound surprising but from a mathematical perceptive the workers are the one who behaved strangely. In mathematics, if you say "I want beef or pork" it can also mean that you want both. It sounds confusing, but it is perfectly logical.
Consider the following: Let P and Q be two statements. When the statement A="P or Q is true" is true?
If P is true and Q is false, A is true.
If P is false and Q is true, A is true.
If P is false and Q is false, A is false.
If P is true and Q is true, A is true.

The first three situations should be obvious. Concerning the forth lets see an example. In the example we will take P="x is positive" and Q="x>1". For x=2 both P and Q are true. Question:
Is 2 positive or greater than 1? The question sounds weird sounds weird, but is it possible to answer "no"? If you answered "no" than it follows that 2 is not positive and not greater than one. But this is not true. Therefore A is true.

We can use the same argument for the "beef or pork" situation. If you say you want beef or pork and are served both, can you say that you didn't get what you ordered?

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