Question

A certain business keeps a database of information about its customers. A. Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning.

Answer 1

If we assume that every person has

(i) a home phone number

(ii) no more than one home phone number

then and only then rule is a function.

Let us denote the people by , where stands for the first name in the book, for the second name, etc. Let the corresponding phone numbers be denoted by , where is the phone number of ,   is the phone number of   and so on.

If for any   and if   , i.e. the phone numbers of and are different then   and are two different people because one person is assumed not to have more than one home phone number.

So, .

Thus, we have   for any . Taking the contrapositive we get for any . So   is a function.

But if any person, say is allowed to have more than one phone numbers, then the value of is not unique. Therefore does not become a function in such a case.

Also if some person does not have a home phone number, then does not exist, and therefore does not become a function in such a case.

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