Question

Challenge: Exactly where is the mistake in the following proof that all personal computers are the same brand? Let \(\mathcal{T}=\\{n \in \mathbb{N}: n \geq 1\) and in every set of \(n\) personal computers, all the personal computers are the same brand \(\\} .\) Prove by induction that for every natural number \(n\) such that \(n \geq 1\) is in \(T\). (Base step) \(1 \in T\), since, trivially, if a set of personal computers contains only one computer, then every (one) computer in the set has the same brand. (Inductive step) Suppose \(n \in T\). We need to show \(n+1 \in T\). So, let \(P\) be any set of \(n+1\) personal computers. Pick any computer \(c \in P ;\) we need to show that every computer in \(P\) is the same brand as \(c\). So, let \(d\) be any computer in \(P\). If \(d=c\), then, trivially, \(d\) and \(c\) are the same brand. Otherwise, \(c \in P-(d\\} .\) The set \(P-(d)\) contains \(n\) computers, so by inductive hypothesis, all the computers in \(P-(d]\) are the same brand. Furthermore, \(d \in P-\mid c\\},\) and. also by inductive hypothesis, all the computers in \(P-\\{c\\}\) are the same brand. Now, let \(e\) be a computer in both \(P-\mid c\\}\) and \(P-[d\\}\). Then, \(d\) is the same brand as \(e,\) and \(c\) is the same brand as \(e\). Therefore, \(d\) is the same brand as \(c\).

Short answer

The mistake is in assuming overlap between subsets \(P-\{d\}\) and \(P-\{c\}\), leading to an erroneous conclusion about equality.

Step-by-step solution

Understand the Base Case

The base case is for the set containing one computer. It states that in a set with one computer, all computers (just this one) must be the same brand. This is trivially true.

State the Inductive Hypothesis

Assume that for any set of \(n\) personal computers, all computers are the same brand. This assumption is for the purpose of proving the statement for \(n+1\) computers.

Form the Set for the Inductive Step

Consider any set \(P\) of \(n+1\) computers. Choose a computer \(c\) from this set. To complete the proof, it must be shown that every computer in \(P\) is the same brand as \(c\).

Analyze the Subsets

Split the set \(P\) into two subsets: \(P-\{d\}\) and \(P-\{c\}\) where \(d\) and \(c\) are different computers. Both subsets have \(n\) computers.

Apply Inductive Hypothesis to Subsets

Apply the inductive hypothesis to both subsets. According to the hypothesis, all computers in \(P\)-\{d\} and all in \(P\)-\{c\} are the same brand within their respective sets.

Identify the Overlap

Note that \(P-\{d\}\) and \(P-\{c\}\) both contain \(n-1\) of the same computers. However, the mistake is claiming any non-empty overlap between \(P-\{d\}\) and \(P-\{c\}\) since all overlap relies on cannot guarantee any intersection; therefore, we can't necessarily conclude that \(c\) and \(d\) are the same brand without assuming some overlap of these subsets which isn't guaranteed.

Key concepts

Base Case

In mathematical induction, the base case is the foundation of the entire proof. It is the first step, where you prove the statement for the smallest value, often starting with 1. Consider a scenario where you want to prove a general statement for a set of personal computers. By beginning with just one computer, you establish that the claim holds true in its most basic form.

• If a set contains only one computer, it is obvious that all computers (which is just one in this case) are of the same brand. This is fundamentally true and helps in kick-starting the induction process.

The base case is crucial because it prevents the assumption from being groundless right at the start. Once you secure the base, you can extend the logic to cover larger sets using the inductive process.

Inductive Hypothesis

The inductive hypothesis is a pivotal part of mathematical induction proofs. It is a strategic step where you assume the statement is true for a generic natural number, say \( n \), and then use this assumption to prove that the statement is also true for \( n+1 \).This step involves:

• Assuming the statement holds for a particular size of the set, such as a set of \( n \) personal computers.
• Using this assumption to help demonstrate that the claim logically extends to a set of \( n+1 \) computers.

This technique creates a domino effect in proofs:

• If we can push down the first domino (the base case), and every subsequent domino causes the next to fall (inductive step), then eventually all dominos (cases) will be down.
• The key here is the logical leap from \( n \) to \( n+1 \), ensuring if it holds for any arbitrary \( n \), the same truth must apply for the next in line.

Set Theory

Set theory is the study of collections of objects, and it plays a critical role in organizing and analyzing data in mathematical proofs. In the context of induction, we often use sets to compare and contrast different groupings of elements.Consider a set \( P \) of \( n+1 \) computers, where we are investigating whether all are of the same brand. Here:

• We split the initial set into two subsets, \( P-\{d\} \) and \( P-\{c\} \).
• Each subset removes one element from the original, leaving \( n \) elements each.

This logical splitting allows us to apply the inductive hypothesis separately to each subset, adhering to our assumptions for smaller sets. Understanding how to manipulate and explore these sets is vital for correctly applying induction in proofs. It ensures we're making valid comparisons and transitions between different cases.

Proof Techniques

Proof techniques are methods mathematicians use to establish truths and validate statements. Mathematical induction, in particular, is one of the most powerful and widely used techniques. Let's delve into various aspects of these techniques:

• Direct Proofs: Establish the truth of a statement outright by logical deductions.
• Indirect Proofs (Contradiction): Assume the opposite of the statement you want to prove, then show this assumption leads to a contradiction.
• Induction: Begin with the simplest case (the base case), assume it's true for an arbitrary case, then prove it for the next step (inductive step).

Specific to induction, the overview goes as follows:

• Start with the base case to set a foundation.
• Form an inductive hypothesis for an arbitrary amount \( n \).
• Complete the inductive step: Prove that if it holds for \( n \), then it holds for \( n+1 \).

Each part of the induction process must be carefully adhered to ensure the proof is valid and applicable to all natural numbers in question. Understanding and mastering these techniques allow for rigorous argumentation and verification in mathematics.