Definition 2.1.1.
A propositional function is a sentence that becomes a statement when ambiguity is removed. We denote a propositional function as \(p(x)\) if it results in a simple statement or \(P(x,y)\) if it results in a compound statement.
Section 2.1 Propositional Functions
Definition 2.1.2.
Given a domain of discourse \(D\text{,}\) a universal quantifier is the specification, “for all \(x\) in \(D\text{,}\)” or, “for each \(x\) in \(D\text{.}\)” It is denoted \(\forall x \in D\text{.}\) The logical shorthand for a universally quantified statement is \(\forall x \in D, p(x)\text{.}\)
Example 2.1.3.
The statement, “For all real numbers \(x\text{,}\) \(x^2 \geq 0\)” is a universally quantified statement with a truth value of true. The statement, “For all real numbers \(x\text{,}\) \(\tan(x)\) is a real number” is a universally quantified statement with a truth value of false.
Definition 2.1.4.
Given a domain of discourse \(D\text{,}\) an existential quantifier is the specification, “there exists \(x\) in \(D\text{,}\)” or, “for some \(x\) in \(D\text{.}\)” It is denoted \(\exists x \in D\text{.}\) An existentially quantified statement takes the form, “there exists \(x\) in \(D\) such that \(p(x)\)” or “\(p(x)\) for some \(x\) in \(D\text{.}\)” The logical shorthand for an existentially quantified statement is \(\exists x \in D \backepsilon p(x)\text{.}\)
Example 2.1.5.
The statement, “There exists a real number \(x\) such that \(x^2 \geq 10\)” is an existentially quantified statement with a truth value of true. The statement, “There exists a real number \(x\) such that \(\cos(x) > 2\)” is an existentially quantified statement with a truth value of false.
Checkpoint 2.1.6.
Are the following quantified statements true or false? Justify your answers.
- “For all real numbers \(x\) and \(y\text{,}\) \(x^2 \geq y+4\text{.}\)”
- “There exist real numbers \(x\) and \(y\) such that \(x^2 \geq y+4\text{.}\)”
- “For all real numbers \(x\text{,}\) there exists a real number \(y\) such that \(x^2 \geq y+4\text{.}\)”
- “There exists a real number \(x\) such that for all real numbers \(y\text{,}\) \(x^2 \geq y+4\text{.}\)”
