(a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C C B uniformly from the power set of B and let Y be its size. Give the joint p.m.f of (X,Y) and compute E(X – Y). Note: X, Y can take value 0 if you pick the empty set. You can either write down...
(3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C C B uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X – Y). Note: X,Y can take value 0 if you pick the empty set. You can either write down a table...
REAL ANALYSIS Question 1 (1.1) Let A be a subset of R which is bounded above. Show that Sup A E A. (1.2) Let S be a subset of a metric space X. Prove that a subset T of S is closed in S if and only if T = SA K for some K which is closed in K. (1.3) Let A and B be two subsets of a metric space X. Recall that A°, the interior of A,...
Let A={1,2,3,4}. Pick a subset B⊆A uniformly among the 2^4 subsets (i.e. the power set ofA) and let X be its size. Then likewise pick a subset C⊆B uniformly from the power set of B and let Y be its size. Give the joint p.m.f of (X, Y) and compute E(X−Y). Hint: X, Y can take value 0 if you pick the empty set. You can either write down a table or a compact expression of the form P(X=i, Y=j).
please 3&4&5 3. Let S = {1,2,...,7,8) be ordered as in figure below. Consider the subset A = {3,6,7} of S. a. Find the set of upper bounds of A b. Find the set of lower bounds of A C. Does sup(A) exist? d. Does inffA) exist? 4. Repeat problem 3 for the subset B = {1,2,4,7) of S. 5. Let S be the ordered set in figure below. Suppose A = {1,2,3,4,5) is order-isomorphic to S and the following...
Let S be the set of outcomes when two distinguishable dice are rolled, let E be the subset of outcomes in which at least one die shows an even number, and let F be the subset of outcomes in which at least one die shows an odd number. List the elements in the given subset. E' (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6) (1, 1), (1, 3), (1, 5),...
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
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...
2. Let A-(2, 3, 4), B = {3, 4, 5, 6), and suppose the universal set is U-(1,2 all the elements in the following sets. ,9). List (a) (A U B) (b) (ANB) × A (c) P(An B)
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...