Special Types of Binary Relations
Some important types of binary relations R between X and Y are listed below.
Uniqueness properties:
- injective (also called left-unique): for all x and z in X and y in Y it holds that if xRy and zRy then x = z.
- functional (also called right-unique or right-definite): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z; such a binary relation is called a partial function.
- one-to-one (also written 1-to-1): injective and functional.
Totality properties:
- left-total: for all x in X there exists a y in Y such that xRy (this property, although sometimes also referred to as total, is different from the definition of total in the next section).
- surjective (also called right-total): for all y in Y there exists an x in X such that xRy.
Uniqueness and totality properties:
- A function: a relation that is functional and left-total.
- A bijection: a one-to-one correspondence; such a relation is a function and is said to be bijective.
Read more about this topic: Binary Relation
Famous quotes containing the words special, types and/or relations:
“The very best reason parents are so special . . . is because we are the holders of a priceless gift, a gift we received from countless generations we never knew, a gift that only we now possess and only we can give to our children. That unique gift, of course, is the gift of ourselves. Whatever we can do to give that gift, and to help others receive it, is worth the challenge of all our human endeavor.”
—Fred Rogers (20th century)
“He types his laboured column—weary drudge!
Senile fudge and solemn:
Spare, editor, to condemn
These dry leaves of his autumn.”
—Robertson Davies (b. 1913)
“Major [William] McKinley visited me. He is on a stumping tour.... I criticized the bloody-shirt course of the canvass. It seems to me to be bad “politics,” and of no use.... It is a stale issue. An increasing number of people are interested in good relations with the South.... Two ways are open to succeed in the South: 1. A division of the white voters. 2. Education of the ignorant. Bloody-shirt utterances prevent division.”
—Rutherford Birchard Hayes (1822–1893)