Discrete structures introduction

1. Discrete Structures
Overview
Lecture Video link:
https://www.youtube.com/watch?v=p4JpkllY_18&list=PLVEVLI2v6thXXsHeo0i48U
uvceypAAL9N

2. Objectives
The course provides a solid theoretical
foundation of discrete structures as they
apply to Computer Science problems and
structures. The students will learn how to use
mathematical notation and solve problems
using mathematical tools.

3. What are “discrete structures”
anyway?
• “Discrete” - Composed of distinct,
separable parts. (Opposite of continuous.)
discrete:continuous :: digital:analog
• “Structures” - Objects built up from simpler
objects according to some definite pattern.
• “Discrete Structure” - The study of
discrete, mathematical (i.e. well-defined
conceptual) objects and structures.

4. Discrete / Continues

5. Why Discrete Structures
➔ First, through this course you can develop your
mathematical maturity: that is, your ability to understand and
create mathematical arguments.
➔ Discrete structure is the gateway to more advanced
courses in all parts of the mathematical sciences.
➔ Discrete structure provides the mathematical foundations
for many computer science courses including data
structures, algorithms, database theory, automata theory,
formal languages, compiler theory, computer security, and
operating systems
➔ Students find these courses much more difficult when they
have not had the appropriate mathematical foundations from
discrete math.

6. Goal of Studying Discrete
Structures
➢ One of the primary goals of this course is
to learn how to attack problems that may be
somewhat different from any you may have
previously seen.
➢ One of my goals as a tutor is to help you
develop the skills needed to master the
mathematical reasoning and problem
solving techniques.

7. Propositional logic, Connectives
and Truth Tables
Lecture Video Link:
https://www.youtube.com/watch?v=e5XyAbA3Lrc&list=PLVEVLI2v6thXXs
Heo0i48UuvceypAAL9N&index=2

8. LOGIC
Logic is the study of the principles and
methods that distinguish between a valid
and an invalid argument.

9. Proposition
• Statement that is true or false.
• A statement is a declarative sentence that is either
true or false but not both. A statement is also
referred to as a proposition
• Examples:
– The capital of New York is Albany.
– The moon is made of green cheese.
– Go to town. (not a proposition – why?)
– What time is it? (not a proposition – why?)
– x + 1 = 2 (not a proposition – why?)

10. Simple/Compound Propositions
• A simple proposition has a value of T/F
• A compound proposition is constructed from
one or more simple propositions using
logical operators
• The truth value of a compound proposition
depends on the truth values of the
constituent propositions

11. Connectives

12. Truth Tables
• A truth table lists ALL possible values of a
(compound) proposition
– one column for each propositional variable
– one column for the compound proposition
– 2n rows for n propositional variables

13. Negation (NOT)
• NOT can be represented by the ~ or 
symbols
• NOT is a logical operator:
p: I am going to town.
~p: I am not going to town.

14. Truth table for ~ (NOT)
p ~p
T F
F T

15. Conjunction (AND)
• The conjunction AND is a logical operator
p: I am going to town.
q: It is raining.
p  q: I am going to town and it is raining.
• Both p and q must be true for the
conjunction to be true.

16. Truth table for ^ (AND)
p q p ^ q
T T T
T F F
F T F
F F F

17. Disjunction (OR)
• Inclusive or - only one proposition needs to
be true for the disjunction to be true.
p: I am going to town.
q: It is raining.
p  q: I am going to town or it is
raining.

18. Truth table for  (OR)
p q p  q
T T T
T F T
F T T
F F F

19. Exclusive OR
• Only one of p and q are true (not both).
p: I am going to town.
q: It is raining.
p  q: Either I am going to town or it is raining.

20. Truth table for  (Exclusive OR)
p q p  q
T T F
T F T
F T T
F F F

21. Conditional statements
• A conditional statement is also called an
implication or an if .. then statement.
• It has the form p  q
p: I am going to town.
q: It is raining.
p  q: If I am going to town, then it is
raining.
• The implication is false only when p is true
and q is false!

22. Truth table for Conditional
statements
p q p  q
T T T
T F F
F T T
F F T

23. Implication - Equivalent Forms
• If p, then q
• p implies q
• If p, q
• q if p
• q whenever p
• p is a sufficient condition for q
• q is a necessary condition for p
• p only if q

24. Converse of an Implication
• Implication: p  q
• Converse: q  p
• Implication:
– If I am going to town, it is raining.
• Converse:
– If it is raining, then I am going to town.

25. Converse of an Implication
p q q  p
T T T
T F T
F T F
F F T
(for p  q, this would be F)
(for p  q, this would be T)

26. Contrapositive of an Implication
• Implication: p  q
• Contrapositive: q  p
• Implication:
– If I am going to town, it is raining.
• Contrapositive:
– If it is not raining, then I am not going to town.
• The contrapositive has the same truth
table as the original implication.

27. Inverse of an Implication
• Implication: p  q
• Inverse: p  q
• Implication:
– If I am going to town, it is raining.
• Inverse:
– If I am not going to town, then it is not raining.
• The inverse of an implication has the same
truth table as the converse of that
implication.

28. Biconditional
• “if and only if”, “iff”
• p  q
• I am going to town if and only if it is
raining.
• Both p and q must have the same truth
value for the assertion to be true.

29. Truth Table for 
(Biconditional)
p q p  q
T T T
T F F
F T F
F F T

30. Truth Table Summary of
Connectives
F
F
T
T
p
F
T
F
T
q pq
pq
pq
p  q
p  q
T
T
F
F
F
F
T
T
T
F
F
F
T
T
F
T
T
F
T
T