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CSC445 , Quiz #2 on Math Preliminaries Instructor: Dr. Hasmik Gharibyan Name_________________________________________ Score _______ Total points: 15 Fill-in the answer of the following questions 1) (2 points: 0.5 point for each question) Given sets A = {0, 4}, B = {1, 2, 3, 4}, C= { n | nN and n>5}and D = {2n+1 | nN} a) A B C D = b) (A B) - D = { 0, 2, 4 } c) B C = { 0, 5 } Note: Complement is taken with respect to N ( N = {0,1,2, . . . } ). d) List all pairs of disjoint sets (give the pair of sets X and Y in the format (X,Y) ) _____________1) A, C____2) A, D_____3) B, C_____________ 2)(0.5 point) Given three sets X, Y, Z. Let X, Y and Z partition N. X = {0, 1, 2, 3, 4, 5} Specify set Z. Y = {n | nN and n>100} Z = { n | nN and 6 n 100} 3) (1.5 points: 0.5 points for each question) Given the function f : NN 10 , if 0 x < 5 f(x) = 2x+5, if 5 x < 10 x, if x 10 Is this function a) Total?(yes/no)__yes__ b)One-to-one?(yes/no)__no___ c)Onto?(yes/no)___no__ 4) (2.5 points: 0.5 point each) a) Given the set X = {0, 1, 2, 3, 4, 5, 6}{ n | nN and n>25}. Check the correct answer: ____ X is a finite set, _V_X is a denumerable set, ___X is an uncountable set b) Given the set X = {0, 1, 2, 3, 4, 5, 6}{ n | nN and 0<n<1000}. Check the correct answer: _V__ X is a finite set, ____X is a denumerable set, ___X is an uncountable set c) Given the set X = {n2+3n -1 | nN }. Check the correct answer: ____ X is a finite set, _V__X is a denumerable set, ___X is an uncountable set d) Given two sets (arbitrary sets): X and Y. XY (X is a proper subset of Y). Is it possible that card(X) = card(Y) ? (yes/no)___Yes____ e) Given sets X and Y. Suppose X = { 2n | nN} and Y = {n+10 | nN} . Check the correct statement (one answer only). _V_ card(X) = card(Y) //explanation: both are denumerable ____ card(X) < card(Y) ____ card(X) card(Y) ____ card(X) > card(Y) ____ card(X) card(Y) 5) (1.5 points, 0.5 point each) a) Given sets X = { 2 } and Y = {4, 6, 8}. Define the set X x Y ( list all elements). X x Y is the Cartesian Product of X and Y sets. X x Y = { [2,4], [2,6], [2,8] } b) Given sets X = and Y = {0, 1, 2, 4}. How many elements does the set X x Y have?(give a number)___0____ c)Given two sets X and Y. Can we say that the set X x Y is a binary relation on X and Y?(yes/no)__yes___ 6) (1 point: 0.5 point each) a) Can the power set of a set be empty?(yes/no)___no____//it contains at least b) Let X={1, 2}. Define the set P(X) (list all elements). P(X) is the power set of X. P(X) = ___{ , {1}, {2}, {1,2} } _____ 7) (1 point) Given the recursive definition of the set X: Basis: 2, 3, 7 X Recursive step: If xX then 2xX. List the 8 smallest elements of X ( list them in ascending order): 2, 3, 4, 6, 7, 8, 12, 14 8) ( 2.5 points: 0.5 point for each question) Given the following graph: a b c d a) Is the graph cyclic?(yes/no)__yes___ b) Is there a path of length 20 in the graph?(yes/no)__yes__ c) How many null-length paths are there in the graph?(give a number) _4_ //one for each node d) Give the in-degree of the node c (give a number)__3____ e) Is the node c adjacent to the node a ?(yes/no)___yes___ 9) ( 2.5 points: 0.5 point for each question) Given the following tree a b e c d f g j k h i l a) The depth of this tree is (give a number)___3____ b) The number of ancestors of the node g is(give a number) ___3___ //every node is its own ancestor c) The number of internal nodes (that are not leavess) is (give a number)___5____ d) Is the given tree a strictly binary tree?(yes/no) ___no_____ e) The frontier of the tree is _______ejcklhi___________