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23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

15 Feb 2016: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

17 Oct 2016: ii) Let α be such that |K(α) : K| is odd. ... and let β = α4.Find. rationals a, b, c such that β = a bα cα2.

14 Oct 2016: c) Let x be a non-zero rational number. Show that there is a rational number y such thatxy = 1. ... d) Show that if a and b are natural numbers such that a is a factor of b then a 6 b.

27 Oct 2016: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R

23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

28 Oct 2016: 7. A map π : E B is called a covering map if there is an open cover {Uα} of B such that π1(Uα)is a disjoint union. ... H(B) H(E) such that π π! is multiplication by N.

21 Nov 2016: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 15. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

26 Oct 2016: 4. (a) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6= ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

14 Oct 2016: c) Let x be a non-zero rational number. Show that there is a rational number y such thatxy = 1. ... d) Show that if a and b are natural numbers such that a is a factor of b then a 6 b.