Problem 4 - Discrete Mathematics with Applications

Question

In Exercises \(1-6,\) a set \(S\) is defined recursively. Find four elements in each case. $$ \begin{array}{l}{\text { i) } 3 \in S} \\ {\text { ii) } x \in S \rightarrow \lg x \in S^{\dagger}}\end{array} $$

Step-by-Step Solution

Answer

The four elements of the set S are \(3, \lg 3, \lg (\lg 3),\) and \(\lg (\lg (\lg 3))\).

Step 1: Find the second element of S

We are given that 3 is an element of S. Now we apply the second condition: since \(3 \in S\), we know that \(\lg 3 \in S\). Thus, \(\lg 3\) is the second element of S.

Step 2: Find the third element of S

We know that \(\lg 3\) is an element of S. Applying the second condition again: since \(\lg 3 \in S\), we know that \(\lg (\lg 3) \in S\). Thus, \(\lg (\lg 3)\) is the third element of S.

Step 3: Find the fourth element of S

We know that \(\lg (\lg 3)\) is an element of S. Applying the second condition one more time: since \(\lg (\lg 3) \in S\), we know that \(\lg (\lg (\lg 3)) \in S\). Thus, \(\lg (\lg (\lg 3))\) is the fourth element of S.

Step 4: List the four elements

Now we have found four elements of the set S: 1. \(3\) 2. \(\lg 3\) 3. \(\lg (\lg 3)\) 4. \(\lg (\lg (\lg 3))\) The four elements of the set S are \(3, \lg 3, \lg (\lg 3),\) and \(\lg (\lg (\lg 3))\).

Key Concepts

Recursive Definition

Recursive definitions are a powerful tool in mathematics which allows for the creation of a sequence of elements where each element is defined based on the preceding ones. The concept originates from the idea of recursion, a fundamental concept in both mathematics and computer science, where a process repeats itself with the aim of reaching a desired outcome or solving a particular problem.

In the context of the exercise, a recursive set is created by starting with an initial element, known as the base case, and then producing new elements following a specific rule or formula. This method is particularly useful in discrete mathematics, where objects are distinct and countable. For instance, in the exercise, starting with the number 3 being in the set S, and using the logarithmic function as a recursive step, successive elements can be generated by applying the logarithm to the previous element.

Discrete Mathematics

Discrete mathematics is an area of mathematics devoted to the study of discrete, rather than continuous, objects. It includes topics such as logic, set theory, combinatorics, graph theory, and algorithms and is often the foundation of computer science and information theory. Discrete objects can be counted and precisely defined, which is why they are the ideal subjects for recursive definitions.

Understanding discrete math is essential for solving the given exercise because it revolves around finding specific elements of a set rather than considering an uninterrupted range of values. The recursive definition used in the exercise, where each subsequent element is derived from the previous one, exemplifies the stepwise nature of discrete mathematics and its application in creating structured sets.

Logarithmic Function

The logarithmic function typically written as \(\log\), is the inverse of exponentiation and plays a significant role in various branches of mathematics, including discrete mathematics. It helps in simplifying expressions involving exponential growth or decay and has properties that make it useful for solving recursive problems, as demonstrated in the exercise.

The function basically answers the question: 'To what exponent must we raise the base to produce a given number?' In the given problem, by successively applying the logarithm, we find the next element in the series. The characteristic that \(\log(ab) = \log(a) + \log(b)\) is particularly useful, as it allows us to break down and analyze the elements produced by the recursive formula. Hence, understanding the fundamental properties of the logarithmic function is crucial to grasp the methodology behind deriving the subsequent elements of the set S as outlined in the exercise.