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In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others.
Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for propositional variables in φ, replacing each occurrence of the same variable by an occurrence of the same formula. For example: ψ: (R → S) & (T → S) is a substitution ...
Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial:
The typical diagram of the definition of a universal morphism. In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them.
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain.
However, in first-order logic, these two sentences may be framed as statements that a certain individual or non-logical object has a property. In this example, both sentences happen to have the common form () for some individual , in the first sentence the value of the variable x is "Socrates", and in the second sentence it is "Plato". Due to ...
The converse of this axiom follows from the substitution property of equality. 2) Axiom Schema of Specification (or Separation or Restricted Comprehension ): If z is a set and ϕ {\displaystyle \phi } is any property which may be satisfied by all, some, or no elements of z , then there exists a subset y of z containing just those elements x in ...
In mathematics, the associative property [1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic , associativity is a valid rule of replacement for expressions in logical proofs .