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e=[(0,1)] Recall that a rational number x = [(a,b)] is positive if ab > 0 in...
2. (a) Prove by structural induction that for all x E {0,1}*, \x = x. (b) Consider the function reverse : {0,1}* + {0,1}* which reverses a binary string, e.g, reverse(01001) = 10010. Give an inductive definition for reverse. (Assume that we defined {0,1}* and concatenation of binary strings as we did in lecture.) (c) Using your inductive definition, prove that for all x, y E {0,1}*, reverse(xy) = reverse(y)reverse(x). (You may assume that concatenation is associative, i.e., for all...
1. Recall that x E R is positive (resp. negative) if x = (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following LIM00 an for a Cauchy sequence n-oo (a) For any E R, exactly one of the following is true: x is positive, is negative, or x= 0 E R is positive if and only if -x is negative. (b) (c) If x, y E R are both positive, then x + y and xy...
1. Suppose that P is the uniform distribution on [0,1). Partition the interval [0,1) into equivalence class such that x ~ y (x is equivalent to y) if x-y є Q, the set of rational numbers 2. Given 1, by the Axiom of Choice, there exists a nonempty set B C [0,1) such that IB contains exactly one member of each equivalence class. Prove each of the following (a) Suppose that q E Qn [0, 1). Show that B (b)...
4. Define f(z) ={z. (Lia z, İftE [0, 1] rational; -z, if x [0,1 irrational 1 f(x) = if z E (, i] rational; Prove that the function f is not integrable on
5. Consider the language L = {1'0/1k e {0,1}* |i >01) >0 Ak = i*j}; to show that Lis! not a regular language using pumping lemma, the correct choice for the word is: a. 10011 x=1- b. 1POP 1P Z=1 Le 1290? 12p* Z=1P OPS 2P Y-101 YEK d. 10P1P y=1" t:P
Let r be any rational number and define L = { x in Q: x < r }, the set of rational numbers less than r. Show that L is a Dedekind cut by proving the following properties: A. There exists a rational number x in L and there exists a rational number y not in L. ( This proves L is nonempty and L is not equal to Q) B. If x in L, then there exists z in...
unique representation in the form -1)*a Every where k E 0,1} and a,b e N with a,b/ 0 and the greatest common divisor of a and b nonzero rational number has a is 1. Use this to show that Q is countable. unique representation in the form -1)*a Every where k E 0,1} and a,b e N with a,b/ 0 and the greatest common divisor of a and b nonzero rational number has a is 1. Use this to show...
Use the well-ordering principle of natural numbers to show that for any positive rational number x ∈ Q, there exists a pair of integers a, b ∈ N such that x = a/b and the only common divisor of a and b is 1.
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
(d) The function f(x)1 is locally integrable on (0, oo). To see whether converges, we consider the improper integrals separately. (The choice of π above is arbitrary.) By considering f (x) lim an show that 11 converges iff p< 1. Next, by considering lim J(z) an -p- dx show that /2 converges iff p +q>1. Finally, combine these results to show that I converges iff p < 1 and p+q1. (d) The function f(x)1 is locally integrable on (0, oo)....