1. SSK3003 DISCRETE STRUCTURES
Prof. Madya Dr.Ali Mamat
Department of Computer Science

2. INTRODUCTION
Discrete mathematics (structures) is the study of mathematical structures (objects) that are fundamentally discrete rather than continuous (vary smoothly)
Continuous objects are studied in calculus.
The term discrete means separate, distinct.
Discrete structures include integers, sets, functions, relations, graphs and trees.
Topics in discrete mathematics include:
Logic
Set theory
Number theory
Proof Techniques (Mathematic Induction)
Counting
Functions and relations

3. INTRODUCTION (cont.) Recursions Graphs and trees33
Discrete mathematics has become popular in recent years because of its application to computer science.
Logic has applications to automated theorem proving (logic programming) and software development.
Set theory → software engineering, databases
Relations → to databases
Proof techniques → analysis of algorithms
Number theory → cryptography
Graphs and trees → data structures

4. CH.1: The Logic of Compound Statements
Learning Outcomes
Evaluate logical statements
Determine two statements are equivalent
Determine the validity of arguments

5. Contents Logical form and logical equivalence Conditional Statements
Valid and invalid arguments
Application: Digital logic circuits

6. 1.1 Logical form and logical equivalence
Logics are formal languages for formalizing reasoning, in particular
for representing information such that conclusion can be drawn.
A logic involves:
A language with a syntax for specifying what is a legal
expression in the language; syntax defines well formed
sentences in the language
Semantics for associating elements of the language with
elements of some subject matter. Semantics defines the
meaning of sentences (link to the domain); i.e., semantics
defines the truth of a sentence with respect to each possible
domain (world). E.g likes(Anas, Azura)
Inference rules for manipulating sentences in the language (e.g modus ponens)

7. Propositional Logic
A statement (or Proposition) is a sentence that is either true or false but not both.
“Two plus two equals four” is a statement
“He is a university student” is not a statement.
Which one is a statement?
√ 2 > 1
1 + 1 = 1
What time is it?
Read this sentence carefully.
x + y > 0

8. Compound Statements A statement can be atomic or compound.
Atomic Statement
Peter hates Lewis : p
It is raining outside: q
√ 2 > 1: r
Peter stays at home : t
Compound statements are formed from existing statements using logical operators, i.e.
¬ (not),
∧ (and),
∨ (or),
→ (imply or if then),
↔ (equivalent if and only if ).

9. Compound Statements
Example
Peter hates Lewis and it is raining outside. : p ∧ q
If it is raining outside then Peter stays at home : q → t
If Peter doesn't hate Lewis then It is raining or √ 2 > 1 :
¬p → (q ∨ r )

10. Truth Table
A truth table displays the relationships among the truth values of statements. In constructing such truth tables, we write “0” (F) for false and “1” (T) for true. p q
pvq
p∧q T F

11. Truth Table

12. Well-Formed Formulas
Statements and compound statements must follow syntax.
Statements that follows syntax are called well-formed formulas (wffs) or formula.
A wff is defined as follows:
Atomic formula is a wff
If p and q are wffs then
¬p, p∨q, p∧q, p→q, p↔q
are wffs

13. Well-Formed Formulas

14. Type of formula A statement/proposition: true or false
Atomic: P, Q, X, Y, …
Unit Formula: P, ~P,
Conjunctive: P ∧ Q, P ∧ ¬Q, …
Disjunctive: P v Q, P v (P∧ X),…
Conditional: P  Q
Biconditional: P ↔ Q

15. Determining truth of a formula
Atomic formulae: given
Compound formulae: via meaning of the connectives
Suppose: P is true Q is false How about: (P v Q)
Truth tables

16. Precedence ∧ ¬ highest v , ↔ lowest
Avoid confusion - use ‘(‘ and ‘)’:
P ∧ Q v X
(P ∧ Q) v X

17. Parenthesizing Parenthesize & build truth tables
Similar to arithmetics:
3*5+7 = (3*5)+7 but NOT 3*(5+7)
A ∧ B v C = (A ∧ B) v C but NOT A ∧ (B v C)
So start with sub-formulae with highest-precedence connectives and work your way out

18. Evaluate The Truth Value
Given a compound statement, how do we compute the truth value?

19. Translating English Sentence
English (even every other human languages) is
ambiguous. E.g. words such as present, “sumbang”
Example “If you are under 4 feet tall and you are younger than 16 years old then you cannot ride the roller coaster."
You can ride the roller coaster : p
You are under 4 feet tall : q
You are younger than 16 years old : r
(q ∧ r ) → ￢p

20. Logical Equivalences
Denition : The propositions p and q are called logically equivalent if they have identical truth values, denoted by p ≡ q.
One way to determine if two propositions are logically equivalent is to use truth table.
The truth table can also be used to show that two propositions are not equivalent.

21. Some Useful Logical Equivalences

22. Logical equivalence (cont.)
Name
p v ¬p ≡ T0
p ∧ ¬p ≡ F0
Inverse laws or negation laws
p v (p ∧ q) ≡ p
p ∧ (p v q) ≡ p
Absorption laws
T0 is a tautology
F0 is a contradiction

23. Suppose someone gives you
Use of Equivalences
Simplification
Suppose someone gives you
~P v (AB) v ~(C v D  H) v P v X↔Y
and asks you to compute it for all possible input values
You can either immediately draw a truth table with 28 = 256 rows
Or you can simplify it first

24. Simplification ~P v (AB) v ~(C v D  H) v P v X↔Y
~P v P v (AB) v ~(C v D  H) v X↔Y
T v (AB) v ~(C v D  H) v X↔Y T
The statement is a tautology – always true…

25. Evaluate Logical Equivalences

26. Tautology
Definition : A tautology (denoted by T0) is a statement that is always true regardless of the truth values of the individual statements.
A contradiction (denoted by F0) is a statement that is always false regardless of the truth values of the individual statements.
A contingency is a statement that is sometimes true and sometimes false.

27. Equivalence & Tautology
Suppose A and B are logically equivalent
Consider proposition (A ≡ B)
What can we say about it?
It is a tautology!
Why?
A B A ≡ B A ↔ B
F F T T
T T T T

28. 1.2 Conditional Statement
Conditional statement (or implication), p → q, is only false when q is false and p is true. E.g.
If you show up for work Monday morning, then you’ll get the job.
Related Implications
The variable p is called the hypothesis (antecedent) and q the conclusion (or consequent).
Caution ! : Only the contrapositive form is logically equivalent to conditional statement. The converse and the inverse are equivalent.

29. Conditional and its contrapositiveq
p → q
¬q → ¬p T F

30. Bi-conditional
The biconditional of p and q is “p if and only if q” and is denoted p ↔ q.
It is true if both p and q have the same truth values and is false if p and q have opposite truth values.
p ↔ q is equivalent to p → q ∧ q → p:
p → q
p is a sufficient condition for q
q is a necessary condition for p. (if q does not occur then p does not occur too, i,e. ¬q → ¬p )

31. Recall Learning Outcomes Evaluate logical statements
Determine two statements are equivalent
Determine the validity of arguments

32. Questions 1. What is an argument?
2. What is a premise and a conclusion?
3. When is an argument valid?
4. How to determine whether an argument is valid.

33. Argument
An argument is a sequence of statements. All the statements before the final one are called premises (or assumption or hypothesis). The final statement is called the conclusion.
An argument is valid whenever the truth of all its premises implies the truth of its conclusion.
How to show a conclusion q logically follows from the premises
Basically, we show that the argument
is tautology.

34. Valid Argument
Example Show that the argument where premises are p v (q v r ) and ¬r , and the conclusion is p v q, is valid.

35. Valid Argument
Example Show that the argument where premises are p v (q v r ) and ¬r , and the conclusion is p v q, is valid.

36. Valid Argument
Example Show that the argument where premises are p v (q v r ) and ¬r , and the conclusion is p v q, is valid.

37. Valid argument p q r q v r p v (q v r ) ¬r p v q pv(qvr) ^¬ r → (pv q)
T T T T F
T T F
T F T
T F F
F T T
F T F
F F T
F F F

38. Logical Implication From the previous table: p1 : pv(qvr) p2: ¬r q1: q
p1 ^ p2 → q1 is a tautology.
If p, q are arbitrary statements such that p → q, then we says p logically implies q, denoted as p  q

39. Rule of Inferences
Although the truth table method always works, however, it is not convenient. Since the appropriate truth table must have 2n lines where n is the number of atomic propositions.
Another way to show an argument is valid is to construct a formal proof. To do the formal proof we use rules of inference.
A rule of inference is a form of argument that is valid.
In rules of inference, premise(s) are written in lines and the conclusion is on the last line preceded with the symbol  which denotes “therefore".
For example the following rule is called Modus Ponens.
In words, if p and p → q are true then q is true.

40. List of Rule of Inferences

41. Recap Introduction to logic Propositions (statement) Logical Operators
wffs
Truth table
Logical equivalences
Tautologies
Conditional statements
Bi-conditional statements
Valid argument
Rule of inferences