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In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...
When the bottom type is uninhabited, a function whose return type is bottom cannot return any value, not even the lone value of a unit type.In such a language, the bottom type may therefore be known as the zero, never or empty type which, in the Curry–Howard correspondence, corresponds to falsity.
The bottom type in type theory, which is the bottom element in the subtype relation. This may coincide with the empty type , which represents absurdum under the Curry–Howard correspondence The "undefined value" in quantum physics interpretations that reject counterfactual definiteness , as in ( r 0 ,⊥)
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is reflexive), then the qualification proper requires the objects to be different. For example, a proper subset of a set S is a subset of S that is different from S, and a proper divisor of a number n is a divisor of n that is different from n.
For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal." ∵ Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ∵ it has no positive integer factors other than itself and one." ∋ 1. Abbreviation of "such that".
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