Search results
Results from the WOW.Com Content Network
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ∅ {\displaystyle \emptyset } for the set that has no member.
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.
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
Consider the two functions f and g mapping from and to natural numbers, defined as follows: To find f(n), first add 5 to n, then multiply by 2. To find g(n), first multiply n by 2, then add 10. These functions are extensionally equal; given the same input, both functions always produce the same value.
Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree. Given two extensions L / K {\displaystyle L/K} and M / L {\displaystyle M/L} , the extension M / K {\displaystyle M/K} is finite if and only if both L / K {\displaystyle L/K} and M / L ...
An extension of A by B is called split if it is equivalent to the trivial extension. There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext 1 R (A, B). [9] The trivial extension corresponds to the zero element of Ext 1 R (A, B).
1. PayPal Honey — Best for finding discount codes. One of the most widely recognized and trusted browser extensions is the Honey browser extension. It stands out for its comprehensive coupon ...
This area also includes one of order theory's most famous open problems, the 1/3–2/3 conjecture, which states that in any finite partially ordered set that is not totally ordered there exists a pair (,) of elements of for which the linear extensions of in which < number between 1/3 and 2/3 of the total number of linear extensions of . [11 ...