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Let P(n) be the proposition that a set with n elements has 2" subsets. What would...
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.
Prove that all sets with n elements have 2n subsets. Countthe empty set ∅ and the whole set as subsets.
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...
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 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...
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S): the power set of a set S; namely, the set of all subsets of S. P*(S): = P(S) - {@}; namely, the set of all non-empty subsets of S. The following question is a challenging one! As a start, may be you try this question for small values of n, say n=1,2,3. Can you make a guess? (1) We all know that P*(On) has...
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
Problem 1 148pts] (1) I 10pts! Let P(n) be the statement that l + 2 + + n n(n + 1) / 2 , for every positive integer n. Answer the following (as part of a proof by (weak) mathematical induction): 1. [2pts] Define the statement P(1) 2. [2pts] Show that P(1 is True, completing the basis step. 3. [4pts] Show that if P(k) is True then P(k+1 is also True for k1, completing the induction step. [2pts] Explain why...
1. What is wrong with the following proof that shows all integers are equal? (Please explain which step in this proof is incorrect and why is it so.) Let P(n) be the proposition that all the numbers in any set of size n are equal. 1) Base case: P(1) is clearly true. 2) Now assume that P(n) is true. That is for any set of size n all the numbers are the same. Consider any set of n + 1...
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that S possesses two dilferent 3-element subsets, the sums of whose elements are equal. (b) Show that S possesses two disjoint subsets, the sums of whose elements are equal
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that...