Related Terms
An irreflexive, or anti-reflexive, relation is the opposite of a reflexive relation: it is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y) on the real numbers. Note that not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not related to themselves (i.e., neither all nor none). For example, the binary relation "the product of x and y is even" is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.
A relation is called quasi-reflexive if every element that is related to some element is related to itself. An example is the relation "has the same limit as" on the set of sequences of real numbers: Not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself.
The reflexive closure of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ is a superset of ~ and ~′ is reflexive on S. This is equivalent to the union of ~ and the identity relation on S. For example, the reflexive closure of x The reflexive reduction of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on S with regard to ~. That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of x≤y is x
Read more about this topic: Reflexive Relation
Famous quotes containing the words related and/or terms:
“Women stand related to beautiful nature around us, and the enamoured youth mixes their form with moon and stars, with woods and waters, and the pomp of summer. They heal us of awkwardness by their words and looks. We observe their intellectual influence on the most serious student. They refine and clear his mind: teach him to put a pleasing method into what is dry and difficult.”
—Ralph Waldo Emerson (1803–1882)
“What had really caused the women’s movement was the additional years of human life. At the turn of the century women’s life expectancy was forty-six; now it was nearly eighty. Our groping sense that we couldn’t live all those years in terms of motherhood alone was “the problem that had no name.” Realizing that it was not some freakish personal fault but our common problem as women had enabled us to take the first steps to change our lives.”
—Betty Friedan (20th century)