The document discusses truth tables and logical connectives such as conjunction, disjunction, negation, implication and biconditionals. It provides examples of truth tables for compound propositions involving multiple variables. De Morgan's laws are explained, which state that the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations. The concepts of tautologies, contradictions and logical equivalence are also covered.

1. Discrete
Structures
Abdur Rehman Usmani
03419019922

2. Truth Table
oThe truth value of the compound proposition depends
only on the truth value of the component propositions.
Such a list is a called a truth table.

3. Example
p q pq pq ¬(pq) (pq) ¬(pq)
T T T T F F
T F T F T T
F T T F T T
F F F F T F
o (pq) ¬(pq)

4. Example
p q r pq ¬r (pq)¬r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
o (p  q)  ¬r

5. Implication (if - then) 
oThe conditional statement p → q is the proposition “if p, then
q.”
oThe conditional statement p → q is false when p is true and q is
false, and true otherwise.
o p is called the hypothesis and q is called the conclusion.

6. Implication (if - then) 
P Q PQ
T T T
T F F
F T T
F F T
p = “You study hard.”
q = “You will get a good grade.”
p → q = “If you study hard, then
you will get a good grade.”

7. Biconditionals (if and Only If) 
p = “Sharif wins the 2012 election.”
q = “Sharif will be prime minister for five years.”
p ↔ q = “If, and only if, Sharif wins the 2012 election, Sharif will
be prime minister for five years.”
p ↔ q does not imply that p
and q are true, or that
either of them causes the other,
or that they have a
common cause.

8. Precedence of Logical Connectives
o~ highest
oɅ second highest
oV third highest
o→ fourth highest
o↔ fifth highest

9. Logical Equivalence
1. 6 is greater than 2
2. 2 is less than 6
two different ways of saying the same thing.
both be true or both be false.
logical form of the statements is important.
p ∧ q is true when, and only when,
q ∧ p is true.
The statement forms are called
logically equivalent

10. Logical Equivalence
oTwo statement forms are called logically equivalent if, and only
if, they have identical truth values for each possible substitution
of statements for their statement variables.
o P ≡ Q.

11. Logical Equivalence
oNegation of the negation of a statement is logically
equivalent to the statement.
o ∼(∼p) ≡ p

12. Logical Equivalence
o∼(p ∧ q) and ∼p ∧ ∼q are not logically equivalent
p =“0 < 1” and let q =“1 < 0.”

13. Logical Equivalence

14. De Morgan’s Laws
oThe negation of the conjunction of two statements is
logically equivalent to the disjunction of their negations.
o ∼(p ∧ q) and ∼p ∨ ∼q are logically equivalent i.e. ∼(p ∧ q) ≡ ∼p ∨ ∼q.

15. De Morgan’s Laws
oNegation of the disjunction of two statements is
logically equivalent to the conjunction of their negations:
qpqp
qpqp


)(
)(

16. De Morgan’s Laws
oWrite negations for each of the following statements:
o John is 6 feet tall and he weighs at least 200 pounds.
o The bus was late or Tom’s watch was slow.
oNegation of these statements
o John is not 6 feet tall or he weighs less than 200 pounds.
o The bus was not late and Tom’s watch was not slow(/“Neither was
the bus late nor was Tom’s watch slow.”)

17. De Morgan’s Laws
o Negation of a disjunction is formed by taking the conjunction of the negations of
the component propositions.
o Negation of a conjunction is formed by taking the disjunction of the negations of
the component propositions.

18. De Morgan’s Laws
o Frequently used in writing computer programs.
o For instance, suppose you want your program to delete all files modified
outside a certain range of dates, say from date 1 through date 2 inclusive.
o ∼(date1 ≤ file_modification_date ≤ date2)
o is equivalent to
o ( file_modification_date < date1) or (date2 < file_modification_date).

19. De Morgan’s Laws

20. Tautologies and Contradictions
A tautology is a statement that is always true.
Examples:
R(R)
(PQ)  (P)( Q)
A contradiction is a statement that is always false.
Examples:
R(R)
((P  Q)  (P)  (Q))
The negation of any tautology is a contradiction, and the negation of any
contradiction is a tautology.

21. Equivalence
Definition: two propositional statements S1 and S2 are said to
be (logically) equivalent, denoted S1  S2 if
They have the same truth table, or
S1  S2 is a tautology
Equivalence can be established by
Constructing truth tables
Using equivalence laws (Table 5 in Section 1.2)

22. Equivalence
Equivalence laws
Identity laws, P  T  P,
Domination laws, P  F  F,
Idempotent laws, P  P  P,
Double negation law,  ( P)  P
Commutative laws, P  Q  Q  P,
Associative laws, P  (Q  R) (P  Q)  R,
Distributive laws, P  (Q  R) (P  Q)  (P  R),
De Morgan’s laws,  (PQ)  ( P)  ( Q)
Law with implication P  Q   P  Q