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- T27. Let A and B be sets, and let f: A → B and g: B –→ A be functions. Suppose that: • go f(a) = a for every a E A; fo g(b) : = b for every b E B. Prove that f is bijective.Suppose a die is cast 10 times. Let X1 be the number of terminations in the set {1, 2}, X2 be the number of terminations in the set {3}, X3 be the number of terminations in the set {4}, and X4 be the number of terminations in the set {5, 6}. (a) Find the joint pmf of X1, X2, X3.(b) Find the joint pmf of X3 and X4.(c) Find the conditional pmf of X2, given that X1 = 2.
- Let X=(x1, X2, X3, X4, X5) and let A be a fuzzy set whose complement is A=((x3, 1), (x4, 1), (X5, 1)) using the complement function C,(a) wheret=0.25. Then Select one: a. HA(X1)0.25 b. None of them C. HA(X1)>0.25, and ua(xa)s0.25 O d. HA(X1)S0.25 and HA(X4) >0.25 e. HA(X1) 20.25 and HA(X4) <0.25Find an x such that (m) + Σkez (m) (45_2m) = k 5-k 0 for any m.1. (a) Let A = {x € R : x > 0 and x? < a}. If sup A = B, prove that B2 = = a. (b) N is the set of positive integers and for each n E N, let An = (1--,1+-). Find (i) N-1An. (ii) U-1 An (c) For each n E N, n! = n(n – 1)(n – 2)... · 2·1. Evaluate |(n + 1)! – (n + 2)!|.
- 1. Find the union C1 U C2 and the intersection C1n C2 of the two sets C1 and C2, where (a) C1 = {0, 1, 2, }, C2 = {2,3, 4}. (b) C1= {x:0 < x <2}, C2= {x :16.3) If f = u + iv is non-constant and analytic in an open set U, which one of the following are analytic as well? (a) g = u - iv, (b) h = v +iu, (c) k = -u -iv, (d) l= iu - v.(a) For two finite sets A and B, prove that AUB≤ A + B. (b) Prove that U1 [Qn (n, n+1)] is countable.1. Let S = (0,7) U 27 +13... Define < on S by a1.49. (!) Let f and g be functions from R to R. For the sum and product of f and g (see Definition 1.25), determine which statements below are true. If true, provide a proof; if false, provide a counterexample. a) If f and g are bounded, then f + g is bounded. b) If f and g are bounded, then fg is bounded. c) If f + g is bounded, then f and g are bounded. d) If fg is bounded, then f and g are bounded. e) If both f + g and fg are bounded, then f and g are bounded. Y13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all positive integers greate than or equal to 2. Define a function T:S → D as follows: For each integer n ≥ 2, T(n) = the set of all even factors of n. a) Find T(10). b) Find T(17) c) Find T(m), where m is any odd positive integer.SEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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