Dense order
Encyclopedia
In mathematics
, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y.
The rational number
s with the ordinary ordering are a densely ordered set in this sense, as are the real number
s. On the other hand, the ordinary ordering on the integer
s is not dense.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y.
The rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s with the ordinary ordering are a densely ordered set in this sense, as are the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. On the other hand, the ordinary ordering on the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is not dense.