A . Prove that Problem 4. (2 points) Let A and B be two sets. Suppose...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
3. Let X and Y be countably infinite sets. (a) Prove: If X and Y are disjoint then XuY is countably infinite. (b) Is the statement in (a) still true if we remove the hypothesis that X and Y are disjoint? If yes, justify your reasoning with a few sentences. If no, provide a counterexample. (P.S. "Counterexample” means that you have to explain why the example you provide demonstrates that the statement is false.)
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection between A and some infinite subset of N.)
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection...
5. Let A and B be compact subsets of R. (a) Prove that AnB is compact (b) Prove that AUB is compact. (c) Find an infinite family An of compact sets for which UAn is not compact. o-f (d) Suppose that An is a compact set for n 21. Prove that An is compact.
1. Let A -(a, b) a, b Q,a b. Prove that A is denumerable. (You may cite any results from the text.) 2. Let SeRnE N) and define f:N-+S by n)- n + *. Since, by definition, S-f(N), it follows that f is onto (a) Show that f is one-to-one (b) Is S denumerable? Explain 3. Either prove or disprove each of the following. (You may cite any results from the text or other results from this assignment.) (a) If...
Problem 2. Let A, B be sets. Prove that if ACB, then P(A) CP(B). Explain why we can conclude that if A= B, then P(A) = P(B). Problem 3. Let A.B be sets. Prove that if P(A) CP(B), then ACB. Explain why we can conclude that if P(A) = P(B), then A= B.
answer question 5 please 3 and 4 are just included to
refer to the theorems
3 Prove the following theorem: Theorem 2.2. Let S be a ser. The following statements are equivalent: (1) S is a countable set, i. e. there exists an injective function :S (2) Either S is the empty ser 6 or there exists a surjective function g: N (3) Either S is a finite set or there exists a bijective function h: N S (4) Prove...
hello sir, solve both questions
Problem 5: Let f : A → B be a function, and let X-A and Y-B. Show that X S(x)) Problem 6: Recall that BA denot es the set of all functions A the function f : P(A) → {0,1}A by B. Fix a set A and defi ne f (X)Xx (the charact erist ic function), VX EP(A) Prove that f is a bijection
Problem 22: Which of the following sets are countable? 1. N × Z 2. Q x Q x Q 3. R x R 4.(pe N p prime 7. Set of all infinite sequences of zeroes and ones.
Problem 22: Which of the following sets are countable? 1. N × Z 2. Q x Q x Q 3. R x R 4.(pe N p prime 7. Set of all infinite sequences of zeroes and ones.
Problem 6 Suppose A and B are countable sets. Prove A × B is a countable set.