prove that the following sets are denumerable N-{5,6} {(x,y)? N x R: xy=1 } Solu
ID: 1888521 • Letter: P
Question
prove that the following sets are denumerableN-{5,6}
{(x,y)? N x R: xy=1 }
Explanation / Answer
First we show that S = {x in N such that x > 6} is denumerable. Just define a function f : S -> N by f(x) = x - 6. It's clearly denumerable. Assume f(x) = f(y). Then x - 6 = y - 6 implying x = y. So f is injective. Let y be in N. Then f(y+6) = (y-6) + 6 = y. So f is surjective. Therefore f is bijective and S is denumerable. Notice that N-{5,6} = S U {1,2,3,4}. Since we just showed S is denumerable and the fact that the union of a denumerable set with a finite set is still denumerable, then N-{5,6} is also denumerable. Let A = {(x,y) in NxR : xy =1}. Define f : N -> A by f(x) = (x,1/x). Assume f(x) = f(y). Then (x,1/x) = (y,1/x) implying x = y. Therefore f is injective. Let (a,b) be in NxR. For ab = 1, b=1/a. Thus (a,b) = (a,1/a) and f(a) = (a,b). Therefore f is surjective. Hence f is bijective and {(x,y) in NxR : xy =1} is denumerable.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.