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An extensional definition gives meaning to a term by specifying its extension, that is, every object that falls under the definition of the term in question.. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class.
Shoenfield states the theorem in the form for a new function name, and constants are the same as functions of zero arguments. In formal systems that admit ordered tuples, extension by multiple constants as shown here can be accomplished by addition of a new constant tuple and the new constant names having the values of elements of the tuple.
Extension (model theory) Extension (proof theory) Extension (predicate logic), the set of tuples of values that satisfy the predicate; Extension (semantics), the set of things to which a property applies; Extension by definitions; Extensional definition, a definition that enumerates every individual a term applies to; Extensionality
In convex geometry and polyhedral combinatorics, the extension complexity of a convex polytope is the smallest number of facets among convex polytopes that have as a projection. In this context, Q {\displaystyle Q} is called an extended formulation of P {\displaystyle P} ; it may have much higher dimension than P {\displaystyle P} .
The extensional definition of function equality, discussed above, is commonly used in mathematics. A similar extensional definition is usually employed for relations: two relations are said to be equal if they have the same extensions.
A field extension L/K is called a simple extension if there exists an element θ in L with L = K ( θ ) . {\displaystyle L=K(\theta ).} This means that every element of L can be expressed as a rational fraction in θ , with coefficients in K ; that is, it is produced from θ and elements of K by the field operations +, −, •, / .
An extension of A by B is called split if it is equivalent to the trivial extension 0 → B → A ⊕ B → A → 0. {\displaystyle 0\to B\to A\oplus B\to A\to 0.} There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext 1
The homotopy extension property is depicted in the following diagram If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map f ~ ∙ {\displaystyle {\tilde {f}}_{\bullet }} which makes the diagram commute.