Question

Prove by contradiction that \(Z\), has no smallest element.

Short answer

The set of integers \( Z \) has no smallest element because for any assumed smallest element \( n \), \( n-1 \) is smaller.

Step-by-step solution

Understand the Assumptions

To prove by contradiction, assume the opposite of what we need to prove. Assume that the set of all integers \( Z \) has a smallest element, say \( n \).

Analyze the Properties of Integers

Recall that for any integer \( n \), the integer \( n-1 \) is also a part of \( Z \). This is because \( Z \) includes all positive and negative whole numbers, including zero.

Establish the Contradiction

If \( n \) is the smallest integer, then \( n-1 \) should not exist in \( Z \) or should be greater than or equal to \( n \). However, \( n-1 < n \), which contradicts our assumption that \( n \) is the smallest element.

Conclude the Proof

Since assuming that \( Z \) has a smallest element leads to a contradiction, our assumption is false. Therefore, the set \( Z \) has no smallest element.

Key concepts

Integers

Integers are the basic building blocks of mathematics. They include all whole numbers and their negative counterparts. This means integers encompass:

• Positive numbers like 1, 2, 3, and so on.
• Negative numbers such as -1, -2, -3, etc.
• Zero (0).

An integer is a number without fraction or decimal and it is often represented by the symbol \( Z \). This symbol originates from the German word 'Zahlen', meaning numbers.

In mathematics, integers are fundamental because they form the basis of many operations and proofs, such as the problem of proving the smallest element of integers through contradiction. Given any integer \( n \), you can always find another integer \( n - 1 \), proving that integers continue indefinitely in both positive and negative directions.

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. A set is typically denoted using curly braces, like \( \{...\} \). In this context, the set \( Z \) represents all integers. Here are some basic concepts of set theory:

• Element: Each object in the set is an element, such as an integer in \( Z \).
• Subset: A set \( A \) is a subset of a set \( B \) if every element of \( A \) is also in \( B \).
• Empty set: A set with no elements, denoted by \( \emptyset \).

Set theory is essential in proving properties of numbers and their relationships. When we assume \( Z \) has a smallest element in a contradiction proof, we're essentially considering subsets of \( Z \) and using their properties to find inconsistencies.

This allows us to understand how integers behave within the larger framework of set theory.

Mathematical Logic

Mathematical logic forms the foundation for making rigorous arguments in mathematics. It involves structures and principles used to ensure that statements and proofs are sound. A key tool in mathematical logic is the proof by contradiction technique, often used in set theory.

To prove a statement by contradiction, you start by assuming the opposite of what you aim to show is true. If this assumption logically leads to a contradiction, then the original statement must be true. Let's break down the proof by contradiction for the smallest integer:

• Assumption: Suppose it is true that \( Z \) has a smallest element.
• Logical Analysis: For any integer \( n \), \( n-1 \) is also an integer, thus contradicting the idea of a smallest integer since \( n-1 < n \).
• Conclusion: The assumption leads to a contradiction. Thus, \( Z \) cannot have a smallest element.

This method of reasoning helps mathematicians strengthen or disprove hypotheses and clarify the nature of integer sets.