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A missing solution is a valid one which is lost during the solution process. Both situations frequently result from performing operations that are not invertible for some or all values of the variables involved, which prevents the chain of logical implications from being bidirectional.
Trivial may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by mathematical induction have two parts: the "base case" which shows that the theorem is true for a particular initial value (such as n = 0 or n = 1), and the inductive step which shows that if the theorem is true for a certain value of n, then it is also true for the ...
One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; [1] "degeneracy" is the condition of being a degenerate case. [2] The definitions of many classes of composite or structured objects often implicitly include inequalities.
There are two cases, depending on the number of linearly dependent equations: either there is just the trivial solution, or there is the trivial solution plus an infinite set of other solutions. Consider the system of linear equations: L i = 0 for 1 ≤ i ≤ M , and variables X 1 , X 2 , ..., X N , where each L i is a weighted sum of the X i s.
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category C op.Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite ...
A functor F : C → D yields an equivalence of categories if and only if it is simultaneously: . full, i.e. for any two objects c 1 and c 2 of C, the map Hom C (c 1,c 2) → Hom D (Fc 1,Fc 2) induced by F is surjective;
In category theory, a branch of mathematics, the opposite category or dual category C op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself.