Section Composites of Functions
In our past mathematical experiences, we have often added and multiplied functions together (e.g., if and map from to then and ). In topology, we generally donβt care about any algebraic structure a set might have, so we will move away from sums and products, and focus on compositions of functions.
The basic idea of function composition is that, when possible, the output of a function is used as the input of a function The resulting function can be referred to as β followed by β and is called the composite of with The notation we use is (note the order β is applied first). For example, if
In this case, the output of the function was used as the input for the function This idea motivates the formal definition of the composition of two functions.
Activity 2.2.
(a)
(b)
(c)
(d)
(e)
In Activity 2.2, we asked questions about whether certain composite functions were injections and/or surjections. In mathematics, it is typical to explore whether certain properties of an object transfer to related objects. In particular, we might want to know whether or not the composite of two injective functions is also an injection. (Of course, we could ask a similar question for surjections.) These questions are explored in the next activity.
Activity 2.3.
(a)
Construct a function that is an injection and a function that is an injection. In this case, is the composite function an injection? Explain.
(b)
Construct a function that is a surjection and a function that is a surjection. In this case, is the composite function a surjection? Explain.
(c)
Construct a function that is a bijection and a function that is a bijection. In this case, is the composite function a bijection? Explain.
In Activity 2.3, we explored some properties of composite functions related to injections, surjections, and bijections. The following theorem summarizes the results that these explorations were intended to illustrate.
Theorem 2.5.
Activity 2.4.
(a)
Prove part (1) of Theorem 2.5.
(b)
Prove part (2) of Theorem 2.5.
(c)
Why is the proof of part (3) of Theorem 2.5 a direct consequence of parts (1) and (2)?