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:

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

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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