Fall 2010
Special Test 1
Topic: Propositional Calculus and Truth Tables
p  (q  r)
(q  s)  t
(p  s)
Prove t using a direct proof.
1.
Given:
2.
Given:
pq
rs
¬(p  s)
Prove (q  ¬r) using proof by contradiction.
Fall 2010
p  (q  r)
rs
¬(q  s)
Prove:
¬p
Fill in the blanks for the following proof.
3.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
4.
Given:
Statement
p  (q  r)
Reason
Given
¬(q  s)
Negation of Conclusion
p
1,5; Modus Ponens (rule 30)
q
6; Commutative law 2b
r
2,9; Modus Ponens (rule 30)
qs
3,11; Conjunction (rule 34)
14; Rule 7b
Use a truth table to prove the following:
(p  q)  (¬p  q)
Fall 2010
Special Test 2
Topic: Logic Circuits
Fall 2010
Special Test 3
Topic: Predicate Calculus
1.
Use DeMorgan's Laws to remove all 's and 's from the following statement.
 [xyz[[(x  y)  (y  z)] (x < z)]]
2.
Fill in the truth table for the following propositions generated
two predicates.
Quantifiers
P(x,y) x  , y  
2
2
x -y=x +y
P(x,y) x,y  {0(F),1(T)}
y  (x  y)
if true y =
if true y =
if true x =
if true x =
xy
xy
xy
yx
yx
xy
by each of the
Fall 2010
3.
Use a truth table to show if the following proposition is true or
x P(x)  x Q(x)  x [P(x)  Q(x)]
false.
Fall 2010
Special Test 4
Topics: Set Theory, Predicate Calculus, & Arithmetic Proofs
1.
Give a predicate calculus definition for each:
AB
AB
A
c
A/B
P(A)
2.
Give the Power Set P(S) for each of the sets below.
a.
S = {,{}}
b.
S = {a,{{ }}}
c.
S = {{{}},{{ }}}
d.
S = {{},{{}}}
Fall 2010
3.
Prove the following using Predicate Calculus.
Given:
Prove:
4.
P  (Q  R)
QS
P  (R  S)
Using arithmetic principles, prove that n
2
+ n - 1 is not divisible
by 3.
Fall 2010
Special Test 5
Topic: Recursion and Second Principle of Induction
1.
For each of the following diagrams indicate whether the diagram is a rooted tree,
an extended binary tree or a full binary tree. If a diagram satisfies the definition for
multiple types of tree list all the types.
Fall 2010
2.
(a) Given the recursive definition of a full binary tree. Be as precise as possible.
(b) State whether the following is a fully binary tree. Based on the definition above,
prove your result by illustrating how the definition is or is not satisfied by the
diagram.
(c) State whether the following is a fully binary tree. Based on the definition above,
prove your result by illustrating how the definition is or is not satisfied by the
diagram.
Fall 2010
3.
Given the following recursively defined sequence:
S0 = 1
Sn = 2Sn-1 + 1
a.
Find first five numbers in the sequence Sn.
b.
Guess at the closed form Sn (i.e. guess at the equation that directly
computes the nth element without using recursion) and prove that your guess is correct by
induction.
Hint: Look at the pattern in part a, and think about
powers of 2!!!
Fall 2010
4. Prove by induction:
5n

i=3n
i = n(8n+4)
for n  1.
Fall 2010
Special Test 6
Topic: Big O and Induction
Fall 2010
Special Test 6-New
(From Homework 5 Practice Problems)
Topic: Integers and Matrices
1.
(a) Define the Fundamental Theorem of Arithmetic. (b) Give two numeric
examples that illustrate the definition.
2.
Given the following zero-one matrices, compute indicated results:
(a)
(b)
(c)
Fall 2010
3.
Define the identity matrix. What properties does it have?
4.
Define the transpose of a matrix and give an example.
5.
Define a symmetric matrix. What is special about a symmetric matrix with respect
to the transpose operation?
Fall 2010
Special Test 7
Topic: Binary Relations
1.
Define the following:
binary relation reflexive irreflexive symmetric antisymmetric transitive equivalence relation partially ordered set -
2.
Determine whether the following relations are reflexive,
irreflexive,
symmetric, antisymmetric, transitive, an
equivalence relation, or partially ordered set.
a.
Define R as xRy  x  y, where x,y  .
b.
Define R as xRy  (x - y) is even, where x,y  .
Fall 2010
3. Let S = {1,2,3,4,5,6} and define a binary relation R, where (x,y) R if and only if y
divides x. Draw a graph diagram for R.
4.
Assume the set of students S = {Fred, Bill, Sam} and courses C = {Chem, Engl,
Math, Latin}
a. Using a graph, draw an example binary relation R on S  C, such that R is not
a function.
b. Using a graph, draw an example binary relation R on S  C, such that R is a
function but not one-to-one.
c. Using a graph, draw an example binary relation R on S  C, such that R is a
one-to-one correspondence.
Fall 2010
Special Test 8
Topic: Counting (Part 1)
1.
a.
An office building contains 27 floors and 37 offices on
each floor. How many offices are in the building?
b.
How many license plates can be made using either two
letters followed by four digits or two digits followed by
letters?
c.
How many positive integers not exceeding 100 are
divisible by 4 or by 6?
2.
a.
How many strings of six letters are there in the English
language?
b.
How many different ways are there to choose a dozen
donuts from the 21 varieties at a donut shop?
four
Fall 2010
3.
a.
In how many ways can a set of two positive integers less
than 100 be chosen?
b.
How many possibilities are there for the first, second, and
third
place finishes in a horse race with 12 horses if all
orders of finish are
possible?
Fall 2010
Special Test 9
Topic: Counting (Part 2)
Special Test 10
Topic: Graph Theory