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Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P). In general, the truth of S says nothing about the truth of its converse, [2] unless the antecedent P and the consequent Q are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal."
The propositional calculus [a] is a branch of logic. [1] It is also called propositional logic, [2] statement logic, [1] sentential calculus, [3] sentential logic, [4] [1] or sometimes zeroth-order logic. [b] [6] [7] [8] Sometimes, it is called first-order propositional logic [9] to contrast it with System F, but it should not be confused with ...
" In this case, unlike the last example, the inverse of the statement is true. The converse is "If a polygon has four sides, then it is a quadrilateral." Again, in this case, unlike the last example, the converse of the statement is true. The negation is "There is at least one quadrilateral that does not have four sides.
This powerful statement is a generalization of the gradient theorem from 1-forms defined on one-dimensional manifolds to differential forms defined on manifolds of arbitrary dimension. The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds.
1 Statement. 2 Proof. 3 Example. 4 One-sided version. ... by the converse of the comparison test, ... From Calculus to Analysis.
Together with a converse operation, this turns Allen's interval algebra into a relation algebra. Using this calculus, given facts can be formalized and then used for automatic reasoning. Relations between intervals are formalized as sets of base relations. The sentences During dinner, Peter reads the newspaper. Afterwards, he goes to bed.
A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function. The converse relation of a function f : X → Y {\displaystyle f:X\to Y} is the relation f − 1 ⊆ Y × X {\displaystyle f^{-1}\subseteq Y\times X} defined by the graph f − 1 = { ( y , x ) ∈ Y × X : y ...
Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.