Problem 4 - A Gentle Introduction to the Art of Mathematics

Question

Write your own sentences containing four quantifiers. One sentence in which the quantifiers appear \((\forall \exists \forall \exists)\) and another in which they appear \((\exists \forall \exists \forall) .\)

Step-by-Step Solution

Answer

For each country, there exists a person such that for each city, there exists a part of the country where the person has lived. There exists a planet such that for every species there, there exists a region where another species can be found.

Step 1: Understand the Quantifiers

Quantifiers are symbols used in logic to indicate the scope of a term in a statement or expression. Common quantifiers include \( \forall \) (for all) and \( \exists \) (there exists).

Step 2: Determine the Order

The problem specifies the order of quantifiers for two sentences: \( \forall \exists \forall \exists \) and \( \exists \forall \exists \forall \). Identify what these orders mean before crafting the sentences.

Step 3: Create Sentence with \( \forall \exists \forall \exists \)

Construct a sentence where the quantifiers appear in the order \( \forall \exists \forall \exists \). For example: 'For every country, there exists a person, such that for every city, there exists a part of the country where the person has lived.' Here, you first consider every country \( \forall \text{country} \), then there exists a person in that country \( \forall \exists \text{person} \), for every city in the country \( \forall \text{city} \), there exists a part of the country \( \exists \text{part} \).

Step 4: Create Sentence with \( \exists \forall \exists \forall \)

Construct a sentence where the quantifiers appear in the order \( \exists \forall \exists \forall \). For example: 'There exists a planet such that for every species living there, there exists a region where another species can be found.' Here, you first state that there exists a planet \( \exists \text{planet} \), for every species on that planet \( \forall \text{species} \), there exists a region \( \exists \text{region} \), such that for another species can be found \( \forall \text{species} \).

Step 5: Review and Revise

Review the constructed sentences to ensure that the quantifiers appear in the correct order and that the sentences make logical sense.

Key Concepts

Universal Quantifier

The universal quantifier, denoted by the symbol \( \forall \), means 'for all' or 'for every'. It is used in logical expressions to indicate that the statement it precedes applies to all elements within a given set. For example, if we say 'For every number \( x \),' we are asserting a property that must be true for all numbers \( x \).

When constructing logical statements using the universal quantifier:\( \forall x, P(x) \), it can be read as 'P(x) is true for all x.' Here, P(x) represents a property or statement about x.

• Example: 'For all dogs, they have four legs' translates to \( \forall x (Dog(x) \rightarrow FourLegs(x)) \).


• Another Example: 'Every student in the class studies hard' can be written as \( \forall x (Student(x) \rightarrow StudiesHard(x)) \).

Existential Quantifier

The existential quantifier, depicted by the symbol \( \exists \), means 'there exists' or 'there is at least one'. This is used in logical expressions to signify that there is at least one element in a given set that satisfies a specific property.

For example, if we state 'There exists a number \( y \),' we are saying that at least one number \( y \) fulfills the given condition.

When using the existential quantifier, \( \exists y, Q(y) \), it reads as 'Q(y) is true for at least one y.' Here, Q(y) represents a property or statement about y.

• Example: 'There exists a cat that is black' translates to \( \exists x (Cat(x) \land Black(x)) \).


• Another Example: 'There is a student who scores above average' can be expressed as \( \exists x (Student(x) \land AboveAverageScore(x)) \).

Logical Expressions

Logical expressions are combinations of symbols that express a logical thought or statement. These expressions typically involve quantifiers (universal and existential), logical connectives (like and, or, not), and variables.

In mathematics and logic, logical expressions are written in a formal language and follow specific syntactic rules, allowing us to rigorously define and manipulate logical statements.

To understand logical expressions:

• Quantifiers: \( \forall \) (universal) and \( \exists \) (existential).
• Variables and Predicates: Variables represent elements of a set, and predicates are properties or statements about those variables.
• Logical Connectives: Symbols such as \( \land \) (and), \( \lor \) (or), \( \eg \) (not), which connect logical statements.

For example, consider the logical expression: 'For every student \( x \), there exists a subject \( y \) such that the student likes the subject.' Written formally, it is \( \forall x \exists y (Student(x) \rightarrow Likes(x, y)) \).

Another example: 'There exists a number \( n \) such that for every number \( m \), \( n \) is greater than \( m \).' Formally, \( \exists n \forall m (n > m) \).