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- If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.2 49) Given nonempty subsets of R², say Y....., let Y = nin enner-xer Fix P E R². For a nonempty set XC R², let v(p.X) = sup[p. xlx € X} Suppose there exists y' € y such that p.y* = v(p.Y"), and for every : € (1,...,n), there exists y, y, such that p.y₁ = v(p.Y.). Then, [Question ID=5892] 1. v(p.Y") < Σj, (P.Y) or (p.Y) > Σ(P.Y,) [Option ID=23562] 2. v(p,Y)= ₁=₁ v(p, Y₂) [Option ID-23563] 3. v(p,Y^) < Σ;=1 v(p.Y) [Option ID=23564] 4. v(p.Y')> ₁ (p.Y₂)By a result of Landau (1953), we know that every tournament has aking (a vertex from which every vertex is reachable by a path of length at most 2). Let T be a tournament such that δ-(T) ≥ 1, that is, d-(v) ≥ 1 for all v ∈ V (T). 1. Show that if x is a king in T, then T has another king in N-(x). 2. Using the answer to the previous question, prove that T has at least 3 kings. 3. For each n ≥ 3, give a construction of a tournament T' with n vertices such that δ-(T') ≥ 1 and T' has exactly 3 kings.
- 2) Take any r > 0 and any c ∈ R. Show that the r-ball Br ( c ) is an open subset of R.Determine if the following is a fundemental set of solutions to y'' -6y' + 25 = 0 Given that y1=e3xcos(4x) and y2= e3xsin(4x)Show on two seperate Venn diagrams : P(A U B^d) = 0.9 P(AU B) = 0.6 Please show values. Thankyou
- (1) If U is a neighborhood of x0 and U ⊂ V, The V is a neighborhood of x0. (2) If U1 ...., U2 are neighborhoods of x0, so is ⋂ni-1U1Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7.(i) Prove that |X ∩ Y| ≥ 4. (ii) Deduce that X ∩ Y ∩ Z is non-empty. [Hint: Consider (X ∩ Y) ∪ Z.]16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.
- Let D be digraph. If id(x) > 0 for all x ∈ V (D), then D contains a circuit.Let E be a measurable subset of (0, 1) and E+y for y E [0, 1) be a translation modulo 1 of E. Taking into account that under above-mentioned assumption the set E+y is measurable and m ( E +y) = mE, construct nonmeasurable set by a suitable choice of E.Consider E={-2+1/n} n=1 to infinity U (3,9) as a subset of R with the usual definition of < (less than) a<b is b-a is postive. Is 0 a lower bound of E?