Converse
In examples 2 and 3 it can be shown that x▷I = I◁x. In example 2 both sides equal the converse x of x, while in example 3 both sides are I when x contains the empty word and 0 otherwise. In the former case x = x. This is impossible for the latter because x▷I retains hardly any information about x. Hence in example 2 we can substitute x for x in x▷I = x = I◁x and cancel (soundly) to give
- x▷I = x = I◁x.
x = x can be proved from these two equations. Tarski's notion of a relation algebra can be defined as a residuated Boolean algebra having an operation x satisfying these two equations.
The cancellation step in the above is not possible for example 3, which therefore is not a relation algebra, x being uniquely determined as x▷I.
Consequences of this axiomatization of converse include x = x, ¬(x) = (¬x), (x∨y) = x∨y, and (x•y) = y•x.
Read more about this topic: Residuated Boolean Algebra
Famous quotes containing the word converse:
“The eyes of men converse as much as their tongues, with the advantage that the ocular dialect needs no dictionary, but is understood all the world over.”
—Ralph Waldo Emerson (1803–1882)
“Lately in converse with a New York alec
About the new school of the pseudo-phallic ...”
—Robert Frost (1874–1963)
“Were you to converse with a king, you ought to be as easy and unembarrassed as with your own valet-de chambre; but yet every look, word, and action should imply the utmost respect.... You must wait till you are spoken to; you must receive, not give, the subject of conversation, and you must even take care that the given subject of such conversation do not lead you into any impropriety.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)