7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in ...
b and c please explian thx i post the question from the book Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E e Fo the set function P by ¡f E is finite, 0, if E is finite 1, if Ec is finite. P(h-10, (b) If is countably infinite, show P is finitely additive but not-additive. (c)...
(3) (10 points) Let H and K be non-empty subsets of a vector space V. The sum of H and K, written H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K: that is H + K = {W EVw = u + v, for some u E H and v EK}. Show that if H and K are subspaces of...
Prove that all sets with n elements have 2n subsets. Countthe empty set ∅ and the whole set as subsets.
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si 3. (20 pts) Let ụ be a finite set, and let...
Prove the theorem Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partition). If the elements of Ω are pairwise disjoint, then ~() is transitive. Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partition). If the elements of Ω are pairwise disjoint, then ~() 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...
Let P(n) be the proposition that a set with n elements has 2" subsets. What would the basis step to prove this proposition PO) is true, because a set with zero elements, the empty set, has exactly 2° = 1 subset, namely, itself. 01 Ploi 2. This is not possible to prove this proposition. 3. po 3p(1) is true, we need to show first what happens a set with 1 element. Because, we can't do P(O), that is not allowed....
Let P(X) be the power set of a non-empty set X. For any two subsets A and B of X, define the relation A B on P(X) to mean that A union B = 0 (the empty set). Justify your answer to each of the following? Isreflexive? Explain. Issymmetric? Explain. Istransitive? Explain.
Let X be a finite set and F a family of subsets of X such that every element of X appears in at least one subset in F. We say that a subset C of F is a set cover for X if X =U SEC S (that is, the union of the sets in C is X). The cardinality of a set cover C is the number of elements in C. (Note that an element of C is a...
A is a finite non-empty set. The domain for relation Ris the power set of A.(Recall that the power set of A is the set of all subsets of A. For X A and Y C A, X is related to Y it X is a proper subsets of Yle, X CY). Select the description that accurately describes relation R. Symmetric and Anti-reflexive Symmetric and Refledve Anti-symmetric and Anti-reflexive Anti-symmetric and Refledive