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.
To determine if a given rule is a function, we need to understand the definition of a function.
A function is a rule that assigns each element from one set, called the domain, to exactly one element in another set, called the codomain. In other words, every input has exactly one output.
Let's analyze each of the given rules:
a) People assigned to their fathers: This rule defines a function because it maps each person to their father, which is another person. Each person has a unique father. Therefore, every input (person) has exactly one output (father).
b) Numbers assigned to the sums of their digits: This rule also defines a function. It takes a number as input and assigns it to the sum of its digits. For example, if we consider the number 123, the sum of its digits (1+2+3) is 6. Each number will have a unique sum of its digits, so every input has exactly one output.
c) Mothers assigned to their children: This rule is not a function, and here's why. A mother can have more than one child. If we consider a mother with multiple children, we cannot assign a unique child to her because she has multiple children. Therefore, this rule does not satisfy the definition of a function.
In summary, rules a) and b) define functions because each input has a unique output. Rule c) does not define a function because a mother can have multiple children, making it impossible to assign a unique child for each mother.