MATH 224 – Discrete Mathematics
Why Study Discrete Math
Determination of the efficiency of algorithms, e.g., insertion sort versus selection
sort. Can you provide an example?
Boolean expressions for controlling loops and conditional statements are based on the
propositional calculus and Boolean algebra. What is an example in C++?
The building blocks of computers – logic gates implement Boolean expressions.
Design of programs and algorithms is similar to developing mathematical proofs.
Conversion of high level languages to machine code makes use of formal language
theory. Name some high level languages. What is machine code?
Induction and recursion are the basis for algorithms and programs that use repeated
instructions, e.g., for statements, while statements and recursive functions.
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MATH 224 – Discrete Mathematics
Why Study Discrete Math Continued
Security including encryption and authentication make use of math concepts. Can
you give an example of where encryption is used? What is authentication?
Many data structures make use of trees, e.g., heaps, binary search trees, databases.
The theory behind caching and paging uses mathematics. Do you know what these
are?
Graph theory is used in communication networks, artificial intelligence, computer
games, computer animation among others.
And much much more.
 Perquisite for CS 340 − Required for ABET accreditation
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MATH 224 – Discrete Mathematics
Propositions
Propositions correspond to Boolean expressions in C++. They are statements,
sometimes represented by variables, that may be either true or false. Examples of
propositions (actually conditional statements) in C++ include:
X <= Y
Z > 10 && Z <= 20
!(Z <= 10 || Z > 20)
Flag
-- where Flag is a boolean variable
!Flag || X % 2 == 0
-- When is X % 2 == 0 ?
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MATH 224 – Discrete Mathematics
Common Logical (connectives) Operators
Note that the conditional is not quite the same as the if statement in C++. In
programming languages the condition is not a proposition but a statement about
what code should be executed based on whether or not a Boolean expression is
either true or false. Can you give an example of a conditional statement in C++?
How can the other operators (→, ↔ and + ) be expressed in C++?
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MATH 224 – Discrete Mathematics
The Meaning of Logical Operators Using Truth Tables
These are called compound propositions or propositional clauses.
Can you create a similar table with one or both of P replaced by P and Q
replaced by Q?
Bitwise operators in C++ among others use & (and) | (or) and ^ (xor). These are
not the same as the logical operators. Read the text on this topic (pages 15 and
16) and be prepared to answer questions.
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MATH 224 – Discrete Mathematics
Using Truth Tables
More
.
complex clauses may be created using multiple truth connectives. In the
truth table below the third and fourth columns show the truth values for two
clauses with a single operator and then the result when these two clauses are
combined with a third operator. What would the truth values be if the left and
right side of the conditional were to be reversed?
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MATH 224 – Discrete Mathematics
Logic Puzzles
Example 18 (pages 13-14): Knights are always truthful and knaves always lie. A
says B is a Knight and B says we are different. If A is a knight then B must also be
a knight. But that leads to a contraction since they must be different. If A is a knave
then B must be a knave since knaves lie. B says they are different which would be a
lie if B is a knave. Thus the two statements are consistent. Both statements are lies.
Problem 43 (page 19): Statements 1 through 100 are: “Exactly n statements are
false.” Which statement must be true? If instead we have, “at least n statements
are false.” which statements are true? What would be the case if there are 99
statements instead of 100? Note that it is possible to have a statement that either
contradicts itself or a group of statements that are contradictory. This is called a
paradox.
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MATH 224 – Discrete Mathematics
More Logic Puzzles
Problem 46 (pages 20): Each member of a group of cannibals either lies or does
not lie. An explorer must determine if a given cannibal tells the truth or not. Why
does asking him if he is a liar not help? What yes or no question should the explorer
ask?
How would you go about analyzing this type of problem? One of the tricks that
often works is the use of a double negative, i.e., the negation of a negation.
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MATH 224 – Discrete Mathematics
More Logic Puzzles
Dr. Who was a British television program from the 1960s and 70s. He was an
explorer who used an old telephone booth as his spaceship. In one episode he had
to choose between two doors. One door led to freedom and the other to his certain
death. There were two robots in the room. One always lied and one always told
the truth. What single yes or no question should Dr. Who ask of one of the robots
in order to determine which door to take?
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MATH 224 – Discrete Mathematics
More Logic Puzzles
Evaluating a While Statement
A jar contains both white and black beans. You take two beans from the jar. If
both of them are black you return one black bean to the jar. If one is black and
one is white you return the white bean. If both are white you take both of them
and add a black bean to the jar. You repeat this step until you are tired or until the
jar has only one bean in it. Will this process ever stop? How many beans will be
left at the end? Can you say anything about the original number of black and
white beans based on the beans that are left? These properties are called
preconditions. Are there any invariant properties?
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MATH 224 – Discrete Mathematics
Logical Equivalences
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MATH 224 – Discrete Mathematics
Semi-formal Proofs
By using propositional clauses that have already been shown to be true it is
possible to derive equivalent propositions or to prove that a proposition is a
tautology (always true). Below is an example of a rather trivial derivation.
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MATH 224 – Discrete Mathematics
Semi-formal Proofs
Here is a more complex derivation.
Using p ↔ q ≡ (p → q) ^ (q → p) show that p ↔ q ≡ (p ^ q) v (¬p ^ ¬q)
p↔q
≡
(p → q) ^ (q → p)
Given
≡
(¬p v q) ^ (¬q v p)
Table 7, Page 25
≡
(¬p ^ ¬q)v(¬p ^ p)v(¬q ^ q)v(q ^ p)
distributive laws
Table 6, Page 24
≡
(¬p ^ ¬q) v F v F v (q ^ p)
negation law, Table 6
≡
(p ^ q) v (¬q ^ ¬p)
identity law, Table 6
≡
(p ^ q) v (¬p ^ ¬q)
commutative law, Table 6
Is (p ↔ q) ↔ [(p ^ q) v (¬p ^ ¬q)] a tautology? Explain your answer.
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