For the indexed family of sets D={Dn : n ∈ N}, where Dn =(2 - 1/n, 4 + 1/n) for n ∈ N, find
a. The union of the family D
b. The intersection of the family D
Exercise 1.3. Let {A,:y er be a family of sets indexed by ſ. a) Show that 4, C U4, Wyel. b) Show that n A, CA,, Vyer. Yel mer
Discrete Mathematics
Problem 2.3. Determine the union and the intersection of each indexed collection (1) Let {An : n E N}, in which An = {-n, n 1} for all n E N = {1, 2, 3,... }. (2) Let B= {Bn :n E N}, in which Bn = [V2, V13) for all n E N.
(3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n=1 An = {0} (b) Un=1 An = [0, 1] (c) n=1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category with properly written domain D and range R such that (a) There exists a subset S of D with f-'[F(S)] + S (b) There exists...
For each n ∈ N, define An = (–2n, 1/n) and A = {An: ∈ N). Find the intersection and the union of the family of set A.
2. Let A = {Aq: a € A} be a family of sets and let B be a set. Prove that (a) Bn UA=U (BOA). αΕΔ QE A (b) Let 4 = {Aq: A E A} and let B = {Beß ef}. Use (a) to write (4) (Uda) (UB) UBR as a union of intersections.
C: Recall that the exponential generating function for the number of de- rangements equals Dn D() 1 x n! (a) Find all poles of D(x) and principal parts at these poles. (b) Use "pole removal" procedure to estimate Dn
C: Recall that the exponential generating function for the number of de- rangements equals Dn D() 1 x n! (a) Find all poles of D(x) and principal parts at these poles. (b) Use "pole removal" procedure to estimate Dn
1. Consider the sets: A = {a, b, c, d, e, f, h, j}, B = {a, b, i }, C = {f, h} and U = {a,b,c,d,e,f,g, h,i,j} a. Draw a Venn diagram and place each element in its appropriate region. Insert a photo of your diagram into your HW document. b. Is C a subset of A? Why? C. Is C a subset of B? Why? d. Is A a subset of B? Why? e. Are B and...
Give an efficient algorithm to compute the union of sets A and B, where n = max(|A|, |B|). The output should be an array of distinct elements that form the union of the sets, such that they appear exactly once in the union. Assume that A and B are unsorted. Give an O(n log n) time algorithm for the problem.
Please help with #7
(7) Prove: The number of derangements of n objects is 1! 2! 3! n(Dn-1+ Dn) which simplifies to where the recursion is given by Dnt1 n+1
(7) Prove: The number of derangements of n objects is 1! 2! 3! n(Dn-1+ Dn) which simplifies to where the recursion is given by Dnt1 n+1
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us