Defining The Ordered Pair Using Set Theory
The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property. This was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954, long after Kuratowski discovered his reduction (below). The Kuratowski definition was added in the second edition of Theory of Sets, published in 1970.
If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below.
Read more about this topic: Ordered Pair
Famous quotes containing the words defining the, defining, ordered, pair, set and/or theory:
“Art, if one employs this term in the broad sense that includes poetry within its realm, is an art of creation laden with ideals, located at the very core of the life of a people, defining the spiritual and moral shape of that life.”
—Ivan Sergeevich Turgenev (18181883)
“The U.S. is becoming an increasingly fatherless society. A generation ago, an American child could reasonably expect to grow up with his or her father. Today an American child can reasonably expect not to. Fatherlessness is now approaching a rough parity with fatherhood as a defining feature of American childhood.”
—David Blankenhorn (20th century)
“Then he rang the bell and ordered a ham sandwich. When the maid placed the plate on the table, he deliberately looked away but as soon as the door had shut, he grabbed the sandwich with both hands, immediately soiled his fingers and chin with the hanging margin of fat and, grunting greedily, began to much.”
—Vladimir Nabokov (18991977)
“Here comes a pair of very strange beasts, which in all
tongues are called fools.”
—William Shakespeare (15641616)
“You have a row of dominoes set up; you knock over the first one, and what will happen to the last one is that it will go over very quickly.”
—Dwight D. Eisenhower (18901969)
“[Anarchism] is the philosophy of the sovereignty of the individual. It is the theory of social harmony. It is the great, surging, living truth that is reconstructing the world, and that will usher in the Dawn.”
—Emma Goldman (18691940)