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23 Nov 2020: 3. The symmetric difference of sets A and B is A B = (A B) (B A). ... 8. Define a relation R on N by setting aRb if a divides b or b divides a.

10 Oct 2020: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.8. Let B be a Borel subset of the interval [0, 1].

30 Nov 2020: B = {z D r > 0, z D(z,r) ,f(z) g(z)}. ... a δ,a δ) i(b ,b ). But a = u(p), so u(W) contains (u(p),u(p)).

2 Dec 2020: ii) If a G then e a = a. (iii) If a,b,b′ G are such that a b = e and a b′ = e, then b = b′. ... A group (G, ,e) is abelian2 if for all a,b G we have a b = b a.

23 Nov 2020: b) Use the Mayer–Vietoris sequence to compute the homology of Σg#Sn, andhence deduce that it is homeomorphic to Sn2g. ... Show that the linear map A : R2 R2 preserves the equivalence relation (a,b) (a′,b′) (aa′,b b′) Z2,and so induces a

14 May 2020: For b R let. ϕ(b) = inf{f(x) : g(x) b, x X}. ... Assuming that ϕ(b) > for all b R, show that ϕ is convex.

4 Feb 2020: a) If f is increasing, then f ′(x) 0 for every x (a,b).(b) If f ′(x) 0 for every x (a,b), then f is increasing.(c) If f ... 12. Suppose f : [a,b] R is differentiable, and that f ′(a) < k < f ′(b).

23 Jan 2020: 5. Let f : B B be a continuous function from the open disc B = {(x,y) R2 :x2 y2 < 1} to itself. ... 11. Suppose f : R R has the intermediate value property that if a,b R andf(a) < c < f(b), then f(y) = c for some y between a and b.

9 Nov 2020: closure of A in L. Show that σ(B) = B for all σ G and that A = BG. ... Q of B which contract to P is finite. [In topological language this says the induced map.

29 Sep 2020: From Riesz representation Theorem, there exists T(x′) Hsuch that. 〈x′,T(x)〉B′B = (x | T(x′))H for all x H. ... AB be a converging sequence, then zn = anbn z and by compactness ofA: aφ(n) a implies bφ(n) za b B since B is closed, and hence z