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In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" [2] or "∃ =1". For example, the formal statement
The Rayo function of a natural number , notated as (), is the smallest number bigger than every finite number with the following property: there is a formula () in the language of first-order set-theory (as presented in the definition of ) with less than symbols and as its only free variable such that: (a) there is a variable assignment assigning to such that ([()],), and (b) for any variable ...
if and only if, iff, xnor propositional logic, Boolean algebra: is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence, depends on the author’s style.
If E is a logical predicate, means that there exists at least one value of x for which E is true. 2. Often used in plain text as an abbreviation of "there exists". ∃! Denotes uniqueness quantification, that is, ! means "there exists exactly one x such that P (is true)".
Kasner used it to illustrate the difference between an unimaginably large number and infinity, and in this role it is sometimes used in teaching mathematics. To put in perspective the size of a googol, the mass of an electron, just under 10 -30 kg, can be compared to the mass of the visible universe, estimated at between 10 50 and 10 60 kg. [ 5 ]
The symbol is used to denote negation. For example, if P(x) is the predicate "x is greater than 0 and less than 1", then, for a domain of discourse X of all natural numbers, the existential quantification "There exists a natural number x which is greater than 0 and less than 1" can be symbolically stated as:
For every natural number x, ... There exists an x such that ... For at least one x, .... Keywords for uniqueness quantification include: For exactly one natural number x, ... There is one and only one x such that .... Further, x may be replaced by a pronoun. For example, For every natural number, its product with 2 equals to its sum with itself.
In the theory of partial orders with one relation symbol ≤, one could define s = t to be an abbreviation for s ≤ t t ≤ s. In set theory with one relation ∈, one may define s = t to be an abbreviation for ∀x (s ∈ x ↔ t ∈ x) ∀x (x ∈ s ↔ x ∈ t). This definition of equality then automatically satisfies the axioms for equality.