Prove that all sets with n elements have 2n subsets. Count
the empty set ∅ and the whole set as subsets.
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Prove that all sets with n elements have 2^n subsets. Count the empty set ∅ and the whole set as subsets.
Use induction to prove that every set of n elements has 2n distinct subsets, for all n ? 0. Hint for the inductive case: fix some element of the set and consider whether it belongs to the subset or not. In either case, reduce to the inductive hypothesis.
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....
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 Ω which belong to at least k of the Ai's show that Pal i-1 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 Ω which belong to at least k of the Ai's show that Pal i-1
4·Let A and B be non-empty subsets of a space X. Prove that A U B is disconnected if A n B)U(A nB) 0. Prove that X is connected if and only if for every pair of non-empty subsets A and B of X such that X A U B we have (A B)U (An B)O.
Prove by induction that if A and B are finite sets, A with n elements and B with m elements, then A x B has nm elements. Also, prove by induction the corresponding results for k sets.
5. Binomial Coefficients (a) How many subsets with at least 5 elements does a set with 8 elements have? n+3 (b). Find the coefficient of z" in (3-2)+ (c). How many ways are there to walk down from the top of Pascal's Triangle and end somewhere on the number 20? 5. Binomial Coefficients (a) How many subsets with at least 5 elements does a set with 8 elements have? n+3 (b). Find the coefficient of z" in (3-2)+ (c). How...
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...
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...
What is the number of subsets of a set with n elements, containing a given element? the textbook answer is 2^(n-1), why do we subtract the given element?
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....