Question

Suppose \(A=\\{1,2,3,4\\}, B=\\{0,1,2\\}, C=\\{1,2,3\\} .\) Let \(f: A \rightarrow B\) be \(f=\\{(1,0),(2,1),\) (3,2),(4,0)\\}\(,\) and \(g: B \rightarrow C\) be \(g=\\{(0,1),(1,1),(2,3)\\} .\) Find \(g \circ f\)

Short answer

The function composition \(g \circ f\) maps the set A to C as follows: \(g \circ f = \{(1,1),(2,1),(3,3),(4,1)\}\)

Step-by-step solution

Understand function composition

Function composition involves applying one function to the results of another. When we have two functions, say 'f' and 'g' and we want to find the composition 'g o f', it means we first apply function 'f', and then apply function 'g' to the results. So, if we have 'f: A -> B' and 'g: B -> C', the composition 'g o f' will map A to C.

Apply function 'f' to A

Using the definition of function 'f', we know that \(f(1) = 0, f(2) = 1, f(3) = 2, f(4) = 0\). This gives us the mapping of set A to set B.

Apply function 'g' to the results

Now we use the mappings from step 2 as inputs for function 'g'. We have \(g(0) = 1, g(1) = 1, g(2) = 3\). Apply these values to the results from step 2. So, the composition \(g \circ f\) gives \(g(f(1)) = g(0) = 1, g(f(2)) = g(1) = 1, g(f(3)) = g(2) = 3, g(f(4)) = g(0) = 1\). Thus, we have the final mapping of set A to set C which completes the composition.

Key concepts

Set Theory

Understanding the concept of sets is fundamental in many areas of mathematics. Set theory is a branch of mathematical logic that studies collections of objects, termed as sets, and their properties. Sets are often denoted by uppercase letters, such as 'A', 'B', and 'C', and the objects contained within a set are called elements. Elements can be anything: numbers, symbols, points in space, other sets, and so on.

In the given exercise, we are dealing with three sets: Set A which includes the numbers 1, 2, 3, and 4; Set B with numbers 0, 1, and 2; and Set C which contains 1, 2, and 3. The nature of these sets is the starting point for discussing functions, as functions describe how each element of one set is paired with an element of another set. It's crucial for students to visualize these sets and understand the role they play in function composition.

Mappings in Mathematics

In mathematics, mappings (or simply functions) are rules that associate every element of a given set, called the domain, with exactly one element of another set, known as the codomain. The terms mappings and functions are often used interchangeably. We represent these associations using ordered pairs, where the first element comes from the domain and the second from the codomain.

In our original exercise, the mapping for function 'f' is provided as a set of ordered pairs, each showing how 'f' connects an element of Set A with an element of Set B. Similarly, function 'g' is represented as a set of ordered pairs mapping elements of Set B to elements of Set C. To emphasize the important characteristics of mappings, remember that each input must have a unique output, which is key to understanding how the composition of these functions will behave.

Composite Functions

Composite functions arise when you combine two (or more) functions to create a new one. This process is analogous to a two-step journey, where you reach an intermediary destination before proceeding to your final stop. In function composition, which is often denoted by \(g \circ f\), you first apply the function 'f' and then apply 'g' to the result of 'f'.

In the context of our exercise, heeding the composition \(g \circ f\) means mapping elements from Set A directly to Set C by successively applying 'f' and then 'g'. To effectively teach the concept of composite functions, it's beneficial to have students work through the individual steps of applying each function, as was illustrated in the solution. Visualizing or even drawing the process can immensely aid in understanding how the elements of the initial set A are transformed step by step until they reach their final associations in set C.