Subject: Field of Topology Only questions 1 and 4 please 1. Show that Q is count
ID: 1720171 • Letter: S
Question
Subject: Field of Topology
Only questions 1 and 4 please
1. Show that Q is countably infinite. 2. Show that the maps f and g of Examples 1 and 2 are bijections. 3. Let X be the two-element set (0, 1). Show there is a bijective correspondence between the set P(Z+) and the cartesian product X°. 4. (a) A real number x is said to be algebraic (over the rationals) if it satisfies some polynomial equation of positive degree with rational coefficients aj. Assuming that each polynomial equation has only finitely many roots, show that the set of algebraic numbers is countable. (b) A real number is said to be transcendental if it is not algebraic. Assuming the reals are uncountable, show that the transcendental numbers are uncount- able. (It is a somewhat surprising fact that only two transcendental numbers are familiar to us: e and . Even proving these two numbers transcendental is highly nontrivial.)
Explanation / Answer
Recall that a set X is countably infinite if it can be bijectively mapped to N (Natural Numbers).
(1) Q is countably infinite.
It suffices to show that P , the set of positive rationals is countably infinite, as Q is the union of {0} , P and {-x: x in P}. (Union of countably infinite sets is countably infinite)
Now any member of P, ie. a positive rational is of the form m/n , where m and n are positive integers with (m,n) =1.
Thus P can be identified as a subset of the set Y points in the plane with integral coordinates.
But the set Y is just the cartesian product of N with itself , hence Y is countable.
Thus we have identified P as a subset of a countably infinite set.
Of course, P is infinite as it contains N as a subset.
Thus Q , the set of rational is countably infinite.
(4) (a) By clearing denominators we may assume that the coefficients of the polynomial are all integers (Z).
As there are only countably infinite number of such polynomials, the result follows.
(b) By definition, the set of transcendental numbers is the complement of the set of algebraic numbers in R. As R is assumed to be uncountable, and we know that the set of algebraic numbers is countbale, the result follows.