
4 (4). Start with a set of 12 numbers: S = {2,4,6,8,10,12,9,16,18,20,22,24} Consider two subsets: A...
3. For the following question, we only consider subsets of the set R of real numbers. In particular, for any set of real numbers S, we have S-R- S For each of the following, write out the resulting set using set-builder notation in the style above i.e., by describing the range(s) of values) (b) GnH (d) GUH
Consider the following functions, where I and J denote two subsets of the set R of real numbers. f: R→R x→1/√(x+1) f(I,J): I→J x→ f(x) (a) What is the domain of definition of f? (b Let y be an element of the codomain of f. Solve the equation f(x)=y in x. Note that you may have to consider different cases, depending on y. (c) What is the range of f? (d) Is f total, surjective, injective, bijective? (e) Find a...
Algorithms
8. The problem 'SET COVER gives two numbers n, k, and a family of n subsets of (,.. n It asks whether it is possible to select k of these subsets such that each number in (1,... ,n) occurs in at least one of the selected subsets. (8.1) Show that the problem 'SET COVER' is in the class NP (8.2) The simplest algorithm to solve set cover just tests all the possible choices of k subsets. How long will...
Let S be the set of all subsets of Z. Define a relation,∼, on S by “two subsets A and B of Z are equivalent,A∼B, if A⊆B.” Prove or disprove each of the following statements: (a)∼is reflexive(b)∼is symmetric(c)∼is transitive
4. Ranking/Unranking Subsets. Let A be a set of n elements and set Sk(A) be the collection of all k-element subsets of A. Recall that |Sk(A)I - (a.) (8 points) Describe a ranking algorithm to rank a k-element subset of an n-element set. (b.) (8 points) Describe an unranking algorithm to unrank an integer 0 < s< [into a ithm to unrank an integer 0 S s <C) k-element subset of an n-element set. (c.) (10 points) As examples, let...
5. Let R denote the set of real numbers. Which of the following subsets of R xR can be written as Ax B for appropriate subsets A, B of R? In case of a positive answer, specify the sets A and B. (a) {(z,y)12z<3, 1<y< 2}, (b) {z,)2+y= 1), (c) {(z,y)|z= 2, y R), (d) {(z,y)|z,yS 0}, (e) {(z,y) z y is an integer).
2. Given the set S-ta,b,c,d,e,f,g,h) a) How many subsets does S have? b) How many subsets have exactly 5 elements? c) A subset is randomly chosen for the collection of all possible a) b) c) subsets. What is the probability that it contains exactly 3 elements? d) A subset is chosen at random from all the subsets. d) What is the probability that it contains the element a?
2. Let S-{a,b,c,d) and let F1, F2 be ơ-algebras of subsets of S2 given by a. Is FînF, a ơ-algebras of subsets of S2? Why (or why not)? b. Is F1 UF, a ơ-algebra of subsets of O? Why(or why not)? c. What is cardinality of 2 ( denoted by #(29) or 12 l). d. Find the Power set of (denoted by 2 ).
2) Show that the set that contains all the subsets of the natural numbers N (i.e. the power set of N usually denoted by 2) is uncountable.
Divide the set on integers into subsets such that any of two members, a and b, of a subset are congruent to each other modulo 4. list the subsets.