Wikipedia:Reference desk/Archives/Mathematics/2006 August 19

cardinality
hi, both F and G are i by j matrix,where i=3 ,j=3 Cij={ (f(x,y),g(x,y))/f(x,y)=i,g(x,y)=j} what is the meaning of this expression


 * I don't know where this comes from, but notationally this is severely challenged, as in confusing abuse of notation. I assume the slash is the bar of set comprehension notation. I also assume that "f" and "g" are the same as "F" and "G". Further, the variables x and y are not bound anywhere; I assume that they are implicitly existentially quantified in the constraint of the set comprehension. After all these preliminaries to make some sense out of the notation, we observe first that, since i=3 and j=3, we can simplify this to:

both F and G are 3 by 3 matrix C33={ (F(x,y),G(x,y))|F(x,y)=3,G(x,y)=3}
 * As the constraint gives as the only possibility for F(x,y) and G(x,y) to be 3, we can further simplify the last line to:

C33={ (3,3)|F(x,y)=3,G(x,y)=3}
 * Note that the values collected in the set are now constant, so this is either a singleton set or empty. The question is, do F and G have an element 3 somewhere at the same position (x,y)? If they do, the constraint can be satisfied, and we get the singleton set

C33={ (3,3) }
 * Otherwise we get the empty set for C33. Somehow I don't think this is what was intended, but that is the closest I can come to assigning a meaning to the jumble of symbols. --Lambiam Talk 08:03, 19 August 2006 (UTC)