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 (1818–1883)
“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 (1818–1883)
“Today everything is different. I can’t even get decent food. Right after I got here I ordered some spaghetti with marinara sauce and I got egg noodles and catsup. I’m an average nobody, I get to live the rest of my life like a schnook.”
—Nicholas Pileggi, U.S. screenwriter, and Martin Scorsese. Henry Hill (Ray Liotta)
“There are two kinds of truth; the truth that lights the way and the truth that warms the heart. The first of these is science, and the second is art.... Without art science would be as useless as a pair of high forceps in the hands of a plumber. Without science art would become a crude mess of folklore and emotional quackery.”
—Raymond Chandler (1888–1959)
“God is the immemorial refuge of the incompetent, the helpless, the miserable. They find not only sanctuary in His arms, but also a kind of superiority, soothing to their macerated egos: He will set them above their betters.”
—H.L. (Henry Lewis)
“The theory of truth is a series of truisms.”
—J.L. (John Langshaw)