Question

For the following formulas, let the universe be \(\mathbb{R}\). Translate each of the following sentences into a formula (using quantifiers): (a) There is no largest number. (b) There is no smallest positive number. (c) Between any two distinct numbers, there is a third number not equal to either of them.

Short answer

(a) \(\forall x \exists y (y > x)\); (b) \(\forall x > 0 \exists y (0 < y < x)\); (c) \(\forall x, y (x \neq y \implies \exists z (x < z < y \lor y < z < x))\)."

Step-by-step solution

Understand the Concept of Quantifiers

Quantifiers such as "for all" (\(\forall\)) and "there exists" (\(\exists\)) are used to express statements about quantities of objects that satisfy certain properties. We will use these to translate the given sentences into mathematical formulas.

Translate Sentence (a)

The sentence "There is no largest number" means that for every number \(x\), it is possible to find another number \(y\) greater than \(x\). This can be written as the formula: \(\forall x \in \mathbb{R}, \exists y \in \mathbb{R} \text{ such that } y > x\).

Translate Sentence (b)

The sentence "There is no smallest positive number" implies that for every positive number \(x\), there exists another positive number \(y\) that is smaller than \(x\) yet still greater than zero. Thus, the formula is: \(\forall x \in \mathbb{R}, (x > 0 \implies \exists y \in \mathbb{R} \text{ such that } 0 < y < x)\).

Translate Sentence (c)

The sentence "Between any two distinct numbers, there is a third number not equal to either of them" can be expressed by saying that for every pair of numbers \(x\) and \(y\) where \(x eq y\), there exists a \(z\) such that \(x < z < y\) or \(y < z < x\). The formula is: \(\forall x, y \in \mathbb{R}, (x eq y \implies \exists z \in \mathbb{R} \text{ such that } (x < z < y) \lor (y < z < x))\).

Key concepts

Real Numbers

In mathematics, real numbers are used to describe a continuum of values, including both rational and irrational numbers. Allowing infinite precision, they fill in all the gaps between the integers. Real numbers are denoted by the symbol \(\mathbb{R}\), representing both positive and negative values, as well as zero.

Real numbers play a crucial role in analysis as well as everyday calculations. They ensure that there are no 'gaps' in the number line, making it seamless. Here are some key points about real numbers:

• Every number on the number line is a real number.
• They can be represented as infinite decimals.
• They include irrational numbers like \(\pi\) and \(\sqrt{2}\), which cannot be expressed as simple fractions.

This concept is critical to understanding problems involving sequences, continuity, and calculus because real numbers allow us to manipulate an entire range of values, not just a countable subset.

Mathematical Logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It addresses the limitations and capabilities of mathematical reasoning. In this context, logic allows us to construct rigorous proofs and develop formulas through logical operators and quantifiers.

Quantifiers, for instance, help to specify the extent of the application of a predicate over a domain, like:\

• \(\forall\) which means "for all," is used when the statement must be true for each element in the domain.
• \(\exists\) which denotes "there exists," indicates that the statement is true for at least one element in the domain.

Mathematical logic is essential to forming the basis for the development of consistent and systematic mathematical theories. By understanding the foundations of logic, students can better understand proofs’ structures and reason through complex problems.

Translation of Sentences into Formulas

The skill of translating sentences into formulas is fundamental in understanding mathematical logic and expressing mathematical thoughts. This involves rephrasing natural language statements into precise mathematical expressions, leveraging logic’s structure.

Through the exercise, the sentences are translated into formulas with steps using quantifiers to perform this translation appropriately.

• In sentence (a), recognizing that no largest number exists means expressing a universal condition that for any given number, another greater one can be found: \(\forall x \in \mathbb{R}, \exists y \in \mathbb{R}\) such that \(y > x\).
• With sentence (b), knowing there is no smallest positive means rephrasing it to say every positive number has something smaller yet still positive: \(\forall x \in \mathbb{R}, (x > 0 \implies \exists y \in \mathbb{R}\) such that \(0 < y < x\)).
• Finally, sentence (c) requires understanding that between distinct numbers, another can exist between them: \(\forall x, y \in \mathbb{R}, (x eq y \implies \exists z \in \mathbb{R}\) such that \((x < z < y) \lor (y < z < x)\)).

Being adept at this translation process is invaluable in more advanced mathematical work, enabling students to quickly convert complex ideas into manageable representations.