Problem 1 - Discrete Mathematics with Applications

Question

In Exercises \(1-6,\) a set \(S\) is defined recursively. Find four elements in each case. $$ \begin{array}{l}{\text { i) } 1 \in S} \\ {\text { ii) } x \in S \rightarrow 2 x \in S}\end{array} $$

Step-by-Step Solution

Answer

The set S is defined recursively as 1 ∈ S and x ∈ S → 2x ∈ S. By following the given rules, we start with 1 and find the next elements by doubling the previous element: \(1, 2, 4, 8\). Therefore, the four elements of the set S are \(1, 2, 4,\) and \(8\).

Step 1: Start with the given element

Since 1 is given as an element of S, we can start with 1 as the first element. 2. If x is an element of S, then 2x is an element of S

Step 2: Find the second element

We know that 1 is an element of S, so according to rule ii), we can multiply it by 2 to get the next element: 2 * 1 = 2. So, 2 is also an element of S. 3. 2 is an element of S, then find another element

Step 3: Find the third element

We know that 2 is an element of S, so according to rule ii), we can multiply it by 2 to get the next element: 2 * 2 = 4. So, 4 is also an element of S. 4. 4 is an element of S, then find another element

Step 4: Find the fourth element

We know that 4 is an element of S, so according to rule ii), we can multiply it by 2 to get the next element: 2 * 4 = 8. So, 8 is also an element of S. So, the four elements of the set S are 1, 2, 4, and 8.

Key Concepts

Recursive Sequences

Recursive sequences are a fundamental concept in mathematics where each term is defined in terms of the previous terms. Think of it like a domino effect where the fall of one domino triggers the fall of the next. Similarly, in a recursive sequence, knowing one (or more) of the initial terms and a rule for determining subsequent terms allows us to build the sequence, piece by piece.

In our exercise, the recursive set is defined with two simple rules: the presence of the initial element, which is 1, and a rule to generate the next element by multiplying the current element by 2. To visualize: we start with 1, then applying the rule gives us 2, and so forth. This sequence is the backbone of many mathematical concepts and can form patterns that lend themselves to easy identification and prediction of future terms. It's a process akin to following a recipe step by step, where each step relies on the completion of the one before it.

Set Theory

Set theory is the mathematical science of understanding and organizing collections of objects, referred to as 'sets'. You can imagine sets like baskets containing certain items, which could range from numbers to letters, or even other sets! The rules of set theory dictate how these baskets can interact, combine, and relate to each other.

The problem given involves defining a set named 'S' recursively, which means that the contents of this 'basket' are determined by a specific set of instructions. In our example, the initial instructions tell us that '1' is in the basket. The next rule tells us that if we have an item 'x' in the basket, we can also include '2x', effectively doubling the item. This mirrors the structure and rules one might find in a game, where each move is determined by the rules of the game and the current state of play. Set theory enables vast areas of mathematics to be discussed in a uniform language and provides tools to deal with infinite and finite collections in an abstract way.

Mathematical Induction

Mathematical induction is a powerful tool, especially when it comes to proving the properties of recursively defined sequences or sets like the one in our exercise. Imagine mathematical induction like climbing a ladder - once you've proven that you can step onto the first rung (the base case) and that stepping onto any rung means you can reach the next one (the inductive step), you can confidently say you can climb as high as the ladder goes.

When applied to our set 'S', induction confirms that starting from 1, and then successively multiplying by 2, will always yield another element in the set. The base case is our starting point '1'. The inductive step shows that if an element 'n' is in the set, so is '2n'. This step-by-step verification moves from certainty in a known fact (the base case) to a general rule that holds true for all terms following the pattern - a testament to the strength and utility of mathematical induction in proving patterns within sequences and sets.