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 a smallest number. (b) Every positive number has a square root. (Do not use the square root symbol; use only multiplication.) (c) Every positive number has a positive square root. (Again, do not use the square root symbol; use only multiplication.)

Short answer

(a) \( \exists c \in \mathbb{R} \) \( \forall x \in \mathbb{R}, \ c \leq x \). (b) \( \forall x > 0, \exists y, \ y^2 = x \). (c) \( \forall x > 0, \exists y > 0, \ y^2 = x \).

Step-by-step solution

Understand the Universe

The universe for these sentences is the set of real numbers, denoted by \( \mathbb{R} \). When dealing with statements about real numbers, the terms might involve concepts like positivity, existence, and relationships between numbers.

Translate Sentence (a)

Sentence (a) states 'There is a smallest number.' In mathematical terms, we need to find a real number, say \( c \), such that \( c \leq x \) for every \( x \) in \( \mathbb{R} \). Using quantifiers, this can be expressed as: \[ \exists c \in \mathbb{R} \) such that \( \forall x \in \mathbb{R}, \ c \leq x. \]

Translate Sentence (b)

Sentence (b) states 'Every positive number has a square root.' This means for every positive real number \( x \), there exists a real number \( y \) such that \( y \times y = x \). Using quantifiers, this is written as: \[ \forall x \in \mathbb{R}, (x > 0) \rightarrow \exists y \in \mathbb{R}, \ y \times y = x. \]

Translate Sentence (c)

Sentence (c) states 'Every positive number has a positive square root.' Similar to sentence (b), but with the condition that the square root \( y \) is positive. In quantifiers, this becomes: \[ \forall x \in \mathbb{R}, (x > 0) \rightarrow \exists y \in \mathbb{R}, \ (y > 0) \land (y \times y = x). \]

Key concepts

Real Numbers

Real numbers, denoted as \( \mathbb{R} \), form the foundation for a large part of mathematics. They include all possible numbers along the number line, creating a complete set that incorporates:

• Integers - numbers like -3, 0, and 5 that have no fractional part.
• Rational numbers - numbers like 1/2 or -4.25 that can be expressed as fractions.
• Irrational numbers - numbers like \( \sqrt{2} \) or \( \pi \) which cannot be written as exact fractions.

Understanding real numbers is critical. They allow us to measure, count, and explore mathematical concepts in the real world and beyond. When you work with problems involving real numbers, everything is unified on a comprehensive number line. This helps us in expressing relationships and mathematical truths easily.
Real numbers also play a vital role when translating sentences with quantifiers into formulas, as seen in statements (a), (b), and (c) from the exercise.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. Let's say we have a number \( x \). If \( y \times y = x \), then \( y \) is a square root of \( x \). Here's what you need to know:

• Every positive number has a square root. Even if it's not a nice integer or rational number, it exists.
• Square roots can be positive or negative. This is because both \( 3^2 = 9 \) and \( (-3)^2 = 9 \) mean \( 3 \) and \( -3 \) are square roots of 9.
• Focus on positive square roots when the problem requests it, as in statement (c) where the positive root is essential.

In exercises with quantifiers, you can find yourself expressing complex ideas like every positive number having a positive square root without directly using the square root symbol.

Existential and Universal Quantifiers

Quantifiers are mathematical symbols that express how many elements of a certain group satisfy a given property:

• Existential quantifier (\( \exists \)) - Indicates that there is at least one element in the set fulfilling the property. For example, in the statement 'There is a smallest number', \( \exists \ c \in \mathbb{R} : \forall x \in \mathbb{R}, \, c \leq x \).
• Universal quantifier (\( \forall \)) - Suggests that every element in the set meets the condition. Like in 'Every positive number has a square root', which translates to: \( \forall x \in \mathbb{R}, \, (x > 0) \rightarrow \exists y \in \mathbb{R}, \, y \times y = x \).

These quantifiers help transform plain language statements into precise mathematical expressions. They are central tools in mathematical logic, allowing us to frame statements clearly for analysis and proofs. Understanding how to use them could significantly enhance your problem-solving skills.