to indicate that is a function. It follows from the definition that two functions are equal if and only if they have the same domain, the same codomain, and the same rule: If and , I say that the rule and the rule are the same if and only if for all . We usually say `` the function " when we mean `` the function ," i.e., we name a function by giving just the name for its rule.

be defined by

Then

If is a proposition form on a set , then determines a function whose
domain is and whose codomain is the set of all propositions.