Organization: University of Massachusetts Dartmouth
References: <18s2mtINN2gf@matt.ksu.ksu.edu>
Date: Sat, 12 Sep 1992 14:29:15 GMT
Lines: 33
In article <18s2mtINN2gf@matt.ksu.ksu.edu> bubai@matt.ksu.ksu.edu (P.Chatterjee) writes:
Hello there,
I was just wondering if somebody on the net could help clarify a few conceptual glitches I have regarding very elementary concepts of set-theory.
a) According to my textbook, the ORDERED PAIR of two objects x and y is the set <x,y> = {{x}, {x,y}}. It also goes on to state that by this definition, the ordered pair is determined by x and y; and the order, x first and y second, is important unless x=y.
What throws me off is the definition; what is the motivation behind it?
Well, you want the definition of <x,y> to be a set from which you can
recover x and y. With this standard definition, you can recover x as the
intersection of the elements in <x,y>. Recovering y seems trickier. If
<x,y> has two elements, then y is their symmetric difference. If it has
one element, then y = x, so you can recover y. (Of course, when I said
above that x is the intersection, I meant that it's the only element of
the intersection.)
b) Let f: X --> Y be an injective mapping. By definition, this means:
for all x, x' in X: f(x) = f(x') ==> x=x'.
Equivalently, x not equal to x' ==> f(x) not equal to f(x').
My question here is: for the firstmentioned implication why is the
CONVERSE not true? (I know it's not but am having trouble finding an
intuitive/logical answerfor it).
But the converse is true. That's part of the definition of 'function'.
It simply says that if you feed the same input into the function twice,
you'll get the same output each time.
--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
