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For example, the guarded quantification For some natural number n, n is even and n is prime. means For some even number n, n is prime. In some mathematical theories, a single domain of discourse fixed in advance is assumed. For example, in Zermelo–Fraenkel set theory, variables range over all sets. In this case, guarded quantifiers can be ...
The term "universe of discourse" generally refers to the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The concept universe of discourse was used for the first time by George Boole (1854) on page 42 of his Laws of Thought ...
For example, in a fiber bundle, the total space is often said to be upstairs, with the base space downstairs. In a fraction, the numerator is occasionally referred to as upstairs and the denominator downstairs, as in "bringing a term upstairs". up to, modulo, mod out by An extension to mathematical discourse of the notions of modular arithmetic.
For example, if the domain of discourse is the set of integers, a function symbol f of arity 2 can be interpreted as the function that gives the sum of its arguments. In other words, the symbol f is associated with the function I ( f ) {\displaystyle I(f)} which, in this interpretation, is addition.
This indicates the importance of the domain of discourse, which specifies which values n can take. [note 1] In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a logical conditional. For example,
For example, if Parent(x, y) denotes that x is a parent of y, then first-order logic cannot express the property that x is an ancestor of y. In second-order logic we can express this by saying that every set of people containing y and closed under the Parent relation contains x: ∀P ((Py ∧ ∀a ∀b ((Pb ∧ Parent(a, b)) → Pa)) → Px).
Almgren–Pitts min-max theory; Approximation theory; Arakelov theory; Asymptotic theory; Automata theory; Bass–Serre theory; Bifurcation theory; Braid theory
Mathematical logic is the study of formal logic within mathematics.Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory).