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The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10} The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line.
A subset A of positive integers has natural density α if the proportion of elements of A among all natural numbers from 1 to n converges to α as n tends to infinity.. More explicitly, if one defines for any natural number n the counting function a(n) as the number of elements of A less than or equal to n, then the natural density of A being α exactly means that [1]
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
As an illustration, let R be the set of real numbers, let Z be the set of integers, let O be the set of odd integers, and let P be the set of current or former U.S. Presidents. Then O is a subset of Z, Z is a subset of R, and (hence) O is a subset of R, where in all cases subset may even be read as proper subset. Not all sets are comparable in ...
A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number x {\displaystyle x} contains at least one other irrational number y ≠ x {\displaystyle y ...
The set E of all finite definitions of real numbers is a subset of A. As A is countable, so is E. Let p be the nth decimal of the nth real number defined by the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal if p is not equal either to 8 or 9, and unity if p is equal to 8 or 9.
A pseudorandom subset of the integers containing the primes as a dense subset. To construct this set, Green and Tao used ideas from Goldston, Pintz, and Yıldırım's work on prime gaps . [ 5 ] Once the pseudorandomness of the set is established, the transference principle may be applied, completing the proof.
If the ω(x, n) are bounded, then ω(x) is finite, and x is called an S number. If the ω(x, n) are finite but unbounded, x is called a T number. x is algebraic if and only if ω(x) = 0. Clearly the Liouville numbers are a subset of the U numbers. William LeVeque in 1953 constructed U numbers of any desired degree. [24]