In everyday conversation we often use language somewhat imprecisely. We can often tell what is meant from the context in which it is said. For example, if you ask a waiter in a restaurant for a cup of coffee then you use the word "cup" very generally and expect anything from a six ounce "cup" to a 12 ounce mug. On the other hand if you are cooking and ask for a cup of sugar then you expect a carefully measured cup.
Effective communication in any field, including mathematics, requires that the speaker and listener agree on the use of words. The purpose of this module is to clarify the way in which we use the word "any." In everyday conversation we might say
As you can see from the two examples above, the word "any" may or may nor may not have the same meaning as the word "every." The question -- "Is anyone here? " -- is completely different then the question -- "Is everyone here?" But the two statements -- "Any fourth-grader knows the name of the first president of the United States" and -- "Every fourth-grader knows the name of the first president of the United States" -- mean exactly the same thing.
Confusion about the word "any" can cause a great deal of difficulty in mathematics. We often use the word "any" with the meaning "every" in the statement of theorems -- for example, we might say
Any integer that ends in a zero is evenly divisible by five.
meaning
Every integer that ends in a zero is evenly divisible by five.
or we might say
For any real number, x, and any real number, y,x + y = y + x
meaning
For every real number, x, and every real number, y,x + y = y + x