Section 2.5 Predicate Logic
Many statements in mathematics can't be modelled using propositional logic. In particular, statements that include words such as “all”, “some”, “no”, etc.
Subsection 2.5.1 Predicate Logic
Definition 2.5.1.
A predicate is a sentence that includes variables, that becomes a statement when specific values are substituted for the variables.
For example, equations involving variables, or inequalities, are predicates.
Predicates are denoted by \(P(x)\text{,}\) where \(P\) represetns the predicate, and \(x\) is the variables. This is called a propositional function. Its input are the values of the variables, and its output is a truth value. For a specific value of \(x\text{,}\) \(P(x)\) becomes a statement which is either true or false.
The domain of a propositional function is the set of all possible values that the input variable can take on.
Proportional functions can be combined with logical operators, such as negation, conjunction, disjunction, etc.
Definition 2.5.2.
Let \(P(x)\) be a predicate, where \(x\) has domain \(D\text{.}\) The truth set of \(P(x)\) is the set of all elements of \(D\) that make \(P(x)\) true when they are substituted for \(x\text{.}\) In other words, the set,