Problem 39 - Discrete Mathematics with Applications

Question

Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z .\) Prove each. $$1_{Y} \circ f=f$$

Step-by-Step Solution

Answer

To prove that \(1_Y \circ f = f\), we need to show that for all \(x \in X\), \(1_Y(f(x)) = f(x)\). By the definition of the identity function, for all \(y \in Y\), \(1_Y(y) = y\). Hence, for \(f(x) \in Y\), we have \(1_Y(f(x)) = f(x)\). Therefore, \(1_Y \circ f = f\).

Step 1: Define the Identity Function

The identity function on a set, denoted as \(1_Y\), is a function that maps every element of the set to itself. In other words, for all \(y \in Y\), \(1_Y(y) = y\).

Step 2: Define the Composition of Functions

The composition of two functions, denoted as \(h \circ g\), is a new function obtained by applying the function g first, and then applying the function h. Mathematically, \((h \circ g)(x) = h(g(x))\).

Step 3: Prove that \(1_Y \circ f = f\)

To prove that \(1_Y \circ f = f\), we need to demonstrate that for all \(x \in X\), \(1_Y(f(x)) = f(x)\). Let \(x \in X\). We have: \((1_Y \circ f)(x) = 1_Y(f(x))\) Now, recall the definition of the identity function: for all \(y \in Y\), \(1_Y(y) = y\). In this case, \(f(x) \in Y\), therefore: \(1_Y(f(x)) = f(x)\) So, we have shown that for all \(x \in X\), \(1_Y(f(x)) = f(x)\), which means \(1_Y \circ f = f\).

Key Concepts

Identity Function

The identity function is a fundamental concept in mathematics. For any set \(Y\), the identity function is denoted by \(1_Y\). This function acts as a mirror for the set, sending each element back to itself. If you think of it as a kind of universal rule for the set \(Y\), it states: "If you give me an element, I'll return it exactly as you gave it." Thus, if you take any element \(y\) in \(Y\), applying \(1_Y\) to \(y\) yields \(1_Y(y) = y\).

Within this structure, the identity function serves an important purpose. It acts as a neutral entity when combined with other functions via composition. This means that if you have any function \(f: X \rightarrow Y\), and you compose it with \(1_Y\), the output doesn’t change; you get \(1_Y \circ f = f\).

• It validates that the identity function exists separately from its interaction with other functions.
• It ensures that the function composition \(1_Y \circ f\) authentically replicates \(f\), maintaining every element's image untouched.

In essence, understanding the identity function forms a baseline for diving into more complex functional and compositional algebra.

Set Theory

Set theory provides the foundations for understanding concepts like the identity function and function composition. A set is a collection of distinct elements or objects, which can also be called a domain or range when we discuss functions. For functions such as \(f: X \rightarrow Y\), \(X\) and \(Y\) are sets.

Fundamentally in set theory, each element in a set can be paired with another element from a different set through a function. This introduces the significant notion that \(f(x)\) for each \(x \in X\) has an image in the set \(Y\). Set theory systematically organizes these elements and functions, laying down the ground rules for how they interact.

• It clarifies how maps or functions pass elements from one set to another.
• It underscores that each element's pairing happens within the boundaries of set relations, adhering to their defined attributes.

Understanding set theory is invaluable for developing deeper insights into mathematical consistency across domains and ranges when composing functions like the identity function.

Proof Writing

Proof writing is the art and craft of demonstrating that certain mathematical statements hold true under the rules of logic. It provides a systematic way of validating the reliability and truth of mathematical assertions.

To prove that \(1_Y \circ f = f\), as outlined in our exercise, involves a series of logical steps:

• We begin by acknowledging our goal: showing that the composition of \(1_Y\) with any function \(f\) doesn’t alter \(f\).


• The next move is defining our terms - knowing what an identity function is, recognizing function composition, and confirming the elements involved are well-defined within their sets \(X\) and \(Y\).


• Finally, the proof is completed by linking these definitions together: because \(1_Y(y) = y\) for all \(y \in Y\), applying this logic shows that \(1_Y(f(x)) = f(x)\). Hence, no change occurs in the composition.

Proof writing requires precision, clarity, and a focused progression of thoughts from assumptions to conclusions. This ensures that the mathematical statements can be consistently replicated and verified.