Following our normal procedure to solve this type of exercise, we start with the definitions.
Definition: “A function f is said to be one-to-one, or an injunction, if and only if f (a) = f (b) implies that a = b for all a and b in the domain of f. A function is said to be injective if it is one-to-one.” Discrete mathematics and its applications By Rosen.
Remark: “We can express that f is one-to-one using quantifiers as ∀a∀b(f (a) = f (b) → a = b) or equivalently ∀a∀b(a≠b → f (a) ≠f (b)), where the universe of discourse is the domain of the function.” Discrete mathematics and its applications By Rosen.
Now, let’s solve the exercise.
Table of Contents
- a) office
- b) assigned bus to chaperone in a group of buses taking students on a field trip
- c) salary
- d) social security number
Consider these functions from the set of teachers in a school. Under what conditions is the function one-to-one if it assigns to a teacher his or her:
a) office
Following the definition of a one-to-one function, we can state that the condition for this function to be one-to-one is that the teacher doesn’t share his/her office. If the office is shared, then it is not one-to-one.
The reason is simple, “A function f is said to be one-to-one, or an injunction, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f”.
If two different teachers share the same office, f(a)=f(b) but a is not equal to b (two different teachers).
b) assigned bus to chaperone in a group of buses taking students on a field trip
The reasoning, in this case, is like the previous one. If only one teacher is assigned per bus, then the function is one-to-one.
c) salary
The most common scenario in salary is that it won’t be one-to-one. Usually, companies have specific salaries for specific positions (like teachers). If at least two teachers have the same salary, then the function is not one-to-one.
Therefore, the answer, in this case, is that this function is one-to-one iff each teacher has a different salary.
d) social security number
As the social security number is unique (like ID number, passport number, etc.) This function is one-to-one.
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