math

Can someone explain this to me.

Determine if the given rule is a function, and if not, explain why.
a. People assigned to their fathers

b. Numbers assigned to the sums fo their digits.

c. Mothers assigned to their children.


a) Here the rule defines a function from the set of all people to itself. The function takes an element of the set (i.e. a person) and maps that person to the person's father (which is, of course, another person and thus also a member of the set).

b) Here the function maps numbers to numbers by taking the sum of the digits.

c) Because a mother can have more than one child you cannot define a function like we did above. It is still possible to define a function by considering mothers with a fixed number, N, of children. The function then maps from the set of all mothers with N children to the N dimensional vector space over the set of all people. The function takes a mother and outputs the N dimensional vector:

[child nr1, child nr. 2,..., child nr. N]

Note that for vectors the ordering of the components matters. This can e.g. be chosen to correspond to the age of the children.

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