Quantifiers are powerful tools in logic, allowing us to make sweeping statements about entire groups. Universal quantifiers (∀) cover all elements, while existential quantifiers (∃) assert the existence of at least one element satisfying a condition.

Understanding variables in quantified statements is crucial. Bound variables are tied to quantifiers, while free variables roam unconstrained. This distinction helps us grasp the scope and meaning of logical expressions in formal reasoning.

Quantifiers and Variables

Universal and Existential Quantifiers

• Universal quantifier (∀) expresses that a predicate is true for all elements in a given domain
• Existential quantifier (∃) expresses that a predicate is true for at least one element in a given domain
• For all (∀) is used to represent the universal quantifier and is read as "for all" or "for every"
  • Example: ∀x P(x) is read as "for all x, P(x) is true"
• There exists (∃) is used to represent the existential quantifier and is read as "there exists" or "for some"
  • Example: ∃x P(x) is read as "there exists an x such that P(x) is true"

Variables in Quantified Statements

• Bound variable is a variable that is quantified by a quantifier within a statement
  • In the statement ∀x P(x), x is a bound variable because it is quantified by the universal quantifier ∀
• Free variable is a variable that is not quantified by any quantifier within a statement
  • In the statement P(x) ∧ Q(y), both x and y are free variables because they are not quantified by any quantifier

Quantified Statements

Components of Quantified Statements

• Quantified statement is a statement that involves one or more quantifiers and a predicate
  • Consists of a quantifier, a variable, and a predicate
  • Example: ∀x (P(x) → Q(x)) is a quantified statement with the universal quantifier ∀, variable x, and the predicate (P(x) → Q(x))
• Predicate is a statement that contains one or more variables and becomes a proposition when the variables are assigned specific values
  • Example: "x is even" is a predicate, and it becomes a proposition when x is assigned a specific value, such as "4 is even"

Domain of Discourse

• Domain of discourse is the set of all possible values that the variables in a quantified statement can take
  • Specifies the context or universe in which the quantified statement is being evaluated
  • Example: In the statement "All students in this class have passed the exam," the domain of discourse is the set of all students in the specific class being referred to
• The domain of discourse can be any non-empty set, such as the set of natural numbers, real numbers, or a specific set of objects
  • Example: In the statement "For every natural number n, n² + n + 41 is prime," the domain of discourse is the set of all natural numbers