Download CSC445 , Quiz #2 on Math Preliminaries

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 | nN and n>5}and D = {2n+1 | nN}
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 | nN and n>100}
Z = { n | nN and 6  n  100}
3) (1.5 points: 0.5 points for each question) Given the function f : NN
 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 | nN 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 | nN 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 | nN }.
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. XY (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 | nN} and Y = {n+10 | nN} .
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 xX then 2xX.
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___________