CISC 1100 – Prof. Zhao–Spring 2011
Practice Final Exam
Relations and Functions
 Know the properties of a relation. Be able to determine if a
relation is reflexive, anti-reflexive, symmetric, anti-symmetric,
transitive or non-transitive.
Problem 1:
Determine each of the relations below defined on set Z (all
integers) is
reflexive, anti-reflexive or neither
symmetric, anti-symmetric or neither
transitive or non-transitive
 R≤: “smaller or equal to”
 Rd: “greater than”
 Ra: “adds up to 6”, e.g., (3,3), (1,5) …
 Know the difference between relations and functions.
Problem 1: Given the relation A = {(5,2), (7,4), (9,10), (11, 20), (x, 5)}
Which of the following values for x will make relation A a function?
A.
B.
C.
D.
7
9
4
11
Problem 2: Which graph represents a function y = f(x)?
(A)
(B)
(C)
(D)
 Know how to evaluate a function, including composed functions.
Problem 1:
Given f (x) = 3x + 7, find f (5), f(a+1).
Problem 2:
Given f(x) = x+2, g(x) = (4x-1)
Compute f (g (2)), g(f(2)), f(f(1)), g(g(1))
Problem 3:
Given
and
find
Problem 4:
and
Given
.
,
find
,
, and
,
.
 Know how to find inverse of a given function
Problem 1:
Find the inverse for the function y = 2x - 12.
Problem 2:
Find the inverse for the function y = 2(x – 12).
Problem 3:
Find the inverse for the function
Problem 4:
Using composition of functions, show that
f (x) = 2x - 3 and g(x) = 0.5x + 1.5 are inverse functions.
Problem 5:
What is value for
A.
B.
C.
D.
25
5
10
Can’t be determined
Counting
 Be fluent with addition principle and multiplication principle.
They are the foundation of all of our counting problems. The
problems are paired up on purpose. Try to feel the difference
between addition and multiplication principles.
Problem 1:
Suppose there are 15 boys and 12 girls in a class. How many ways
can you select one person to check attendance?
Problem 2:
Suppose there are 15 boys and 12 girls in a class. We need to select
one boy and one girl to host a performance show. How many
possible selections are there?
Problem 3:
A man has 3 different suits, 4 different shirts and 5 different pairs
of shoes. In how many different ways can this man wear a suit, a
shirt and a pair of shoes?
Problem 4:
A man has 3 white short sleeve shirts, 4 blue short sleeve shirts,
and 5 long sleeve shirts. In how many different ways can he
choose to wear a shirt?
Problem 5:
A person is traveling from city A to city B. There are 8 ways to
take buses, 5 ways to drive, and 2 ways to take a train, 1 way to
fly. How many choices does he have to travel from A to B.
Probl em 6:
A person is traveling from city A to city C. But there is no direct
route from A to C. He must first get to city B, then to city C. There
are 8 ways to go from A to B, 5 ways to go from B to C. How
many choices does he have to travel from A to C.
 Know how to compute P(n, r), C(n, r).
 Know how to use permutation and combination to solve two
special kinds of counting problems (i.e. reuse objects? Order or
objects?). The problems are paired up on purpose. Try to feel the
difference of permutation problems and combination problems.
Problem 1:
Compute P(10, 5), P(59, 2), P(8,8)
Problem 2:
Compute C(10, 5), C(59, 2), C(8,8)
Problem 3:
You have 5 book and you want to put 3 of them onto the shelf. In
how many different ways could you arrange them?
Problem 4:
You have 5 book and you want to put 3 of them onto the shelf. In
how many different ways could you arrange them?
Problem 5:
You have 5 books, you want to choose 3 of them to read. In how
many different ways could you choose?
Problem 6:
How many ways can you pick 3 dessert out of a menu of 10?
Problem 7:
How many ways can a resturant offer 3 favorite dessert with order
of preference out of a menu of 10?
Problem 8:
How many distinguishable ways are there to arrange the letters in
the word “SQUARE”?
 Be able to use combined counting skills to solve more complicated
problems.
Problem 1:
Assuming that any arrangement of letters forms a 'word', how
many 'words' of any length can be formed from the letters of the
word SQUARE? (No repeating of letters)
Problem 2:
How many distinguishable ways are there to arrange the letters in
the word “ MISSISSIPPI ”?
Probability
 Use definition of probability and counting techniques to solve
problems.
Problem 1:
If you flip a fair coin 3 times,
What’s the probability of getting exactly one head?
What’s the probability of getting exactly two heads?
What’s the probability of getting at least one head?
Problem 2:
A die is rolled; find the probability that the number obtained is
greater than 4.
 Understand disjoint and independent events. Use addition rule
and multiplication rule to solve problems.
Problem 1:
A box contains 8 red marbles, 4 white marbles, and 12 black
marbles. A marble is drawn at random, what is the chance of
getting a red or black marble?
Problem 2:
Carlos plays college soccer. He makes a goal 65% of the time he
shoots. If Carlos is going to attempt two goals in the next game.
What’s the probability Carlos makes one of the goals?
What’s the probability Carlos makes both goals?
What’s the probability Carlos makes at least one goal?
What’s the probability Carlos makes no goals?
Problem 3:
You have one standard 6-sided die. You roll it 3 times and record
the results.
How many possible outcomes are there?
What is the probability that you get all 6’s?
What is the probability that you get exactly two 6’s?
 Understand the concept of expected value. Be able to determine a
fair price for a probability game.
Problem 1:
500 tickets for prizes are sold for $2 each. Five prizes will be
awarded – one for $300, one for $200, and three for $50. Steven
purchases one of the tickets. What is the expected value of his
ticket?
Algorithms
 Understand the difference between BubbleSort and MergeSort,
Linear Search and Binary Search. Be able to following the
execution of an algorithm.
Problem 1:
We want to sort (3, 5, 1, 9, 2) into ascending order
A)
How many comparisons are there if we use the following
BubbleSort algorithm? Show your steps.
B)
If we use the following MergeSort algorithm, how many times the
function merge(l1; l2) will be invoked?
BubbleSort Algorithm
Repeat as i varies from n down to 2
Repeat as j varies from 1 to i - 1
3 If lj > lj+1 swap lj with lj+1
1
2
Note. First Repeat loop spans both line 2 &3
MergeSort Algorithm
function mergesort(L)
1 if L has one element then return(L); otherwise continue
2 l1 = mergesort(left half of L)
3 l2 = mergesort(right half of L)
4 L = merge(l1; l2)
5 return(L)
G
Problem 2:
7 positive integers are stored in an ascending order:
2 5 10 15 40
48 60
If we want to write an algorithm to locate the position of a given number,
e.g. position(5) = 2


What searching algorithm would you choose?
Give an example of the worst case scenario of your algorithm and specify
how many comparisons it needs to locate the number. Show your work.