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The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
In predicate logic, a predicate P over some domain is called decidable if for every x in the domain, either P(x) holds, or the negation of P(x) holds. This is not trivially true constructively. Markov's principle then states: For a decidable predicate P over the natural numbers, if P cannot be false for all natural numbers n, then it is true ...
Interesting number paradox: The first number that can be considered "dull" rather than "interesting" becomes interesting because of that fact. Potato paradox : If potatoes consisting of 99% water dry until they are 98% water, they lose 50% of their weight.
In this context, it is worth noting that the probability also approaches one that a random -vertex graph has Hadwiger number greater than or equal to its chromatic number, so the Hadwiger conjecture holds for random graphs with high probability; more precisely, the Hadwiger number is with high probability proportional to / .
In particular, if the Erdős–Straus conjecture itself (the case =) is false, then the number of counterexamples grows only sublinearly. Even more strongly, for any fixed k {\displaystyle k} , only a sublinear number of values of n {\displaystyle n} need more than two terms in their Egyptian fraction expansions. [ 17 ]
The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is ...
Suppose a digraph G has the property that, for every two vertices u and v, either there is an edge from u to v or the outdegree of u plus the indegree of v equals or exceeds the number of vertices in G. Then, according to Woodall's theorem, G contains a directed Hamiltonian cycle. Ore's theorem may be obtained from Woodall by replacing every ...
These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: "For any integer x, if x > 5 then x > 3." [11] – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 ...