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23 Mar 2007: Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic ... all supersets of B.

18 Nov 2007: GROUPS EXAMPLES 4. G.P. Paternain Michaelmas 2007. The questions on this sheet are not all equally difficult and the harder ones are marked with ’s. ... 9. Let X be the set of all 2 2 real matrices with trace zero.

20 Oct 2007: Show, directly from the definition of holomorphic maps, that if f is holomorphic onR {p} then f is in fact holomorphic on all of R.(ii) Suppose that each of A = { ... algebraically related’: there is a polynomial Q in two variables, so that Q(f(z),g(z))

6 Oct 2007: Whatcan you say if n is even? Find all the homomorphisms from Cn to Cm. ... If we think of the planeas C you may assume that all isometries have the form z 7 az b or z 7 az̄ b, where a and b arecomplex numbers

3 Nov 2007: Let G be the set of all 3 3 matrices of the form 1 x y0 1 z. ... If we think of the plane as C you may assumethat all isometries have the form z 7 az b or z 7 az̄ b, where a and b are complex

19 Nov 2007: GROUPS EXAMPLES 4. G.P. Paternain Michaelmas 2007. The questions on this sheet are not all equally difficult and the harder ones are marked with ’s. ... 9. Let X be the set of all 2 2 real matrices with trace zero.

10 Oct 2007: Whatcan you say if n is even? Find all the homomorphisms from Cn to Cm. ... If we think of the plane. as C you may assume that all isometries have the form z 7 az b or z 7 az̄ b, where a and b arecomplex

6 Nov 2007: 7. Let G be the set of all 3 3 matrices of the form. ... If we think of the plane as C you may assumethat all isometries have the form z 7 az b or z 7 az̄ b, where a and b are complex

25 Oct 2007: b) Let R([0, 1]. )denote the vector space of all integrable functions on [0, 1]. ... 10. (a) Show that the space is complete. Show also that c0 = {x : xn 0}, the vector subspace of consisting of all sequences converging to 0, is complete.

17 Oct 2007: 6. Solve (i.e., find all solutions of) the following congruences:(a) 7x 77 (mod 40).(b) 12y 30 (mod 54).(c) 3z 2 (mod 17) and 4z 3 (mod 19). ... 13. Recall the Fibonacci numbers Fn from question 5.(a) Prove that (Fn,Fn1) = 1 for all n.(b) Let m 3.