Introduction to DSA

1. Discrete Structures
BSCS (3 Credit Hour)
Lecture # 02

2. Truth tables
 Truth tables for:
~ p  q
~ p  (q  ~ r)
(p  q)  ~ (p  q)

3. Truth table for the statement form ~ p  q
p q ~p ~ p  q
T T F F
T F F F
F T T T
F F T F

4. Truth table for ~ p  (q  ~ r)
p q r ~ r q  ~
r
~ p ~ p  (q  ~
r)
T T T F T F F
T T F T T F F
T F T F F F F
T F F T T F F
F T T F T T T
F T F T T T T
F F T F F T F
F F F T T T T

5. Truth table for (p  q)  ~ (p  q)
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

6. Double Negative Property ~(~p)  p
p ~p ~(~p)
T F T
F T F

7. Example:
 “It is not true that I am not happy”
 Prove it double negation property for the above
statement ~ (~p)  p
 Solution:
Let p = “I am happy”
Then ~p = “I am not happy”
and ~(~ p) = “It is not true that I am not happy”
Since ~ (~p)  p
Hence the given statement is equivalent to:
“I am happy”

8. ~ (p  q) and ~p  ~q are not logically
equivalent
p q ~p ~q p 
q
~(pq
)
~p  ~q
T T F F T F F
T F F T F T F
F T T F F T F
F F T T F T T

9. DE MORGAN’S LAWS:
 The negation of an and statement is logically equivalent
to the or statement in which each component is negated.
Symbolically: ~(p  q)  ~p  ~q.
 The negation of an or statement is logically equivalent to
the and statement in which each component is negated.
Symbolically: ~(p  q)  ~p  ~q.

10. Truth Table for ~(p  q)  ~p  ~q
p q ~p ~q p 
q
~(p  q) ~p 
~q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T

11. Application of De morgan’s Law using
statements:
 Give negations for each of the following
statements:
The fan is slow or it is very hot.
Akram is unfit and Saleem is injured.
 Solution:
The fan is not slow and it is not very hot.
Akram is not unfit or Saleem is not injured.

12. Exercise:
Associative Law
 Are the statements (p  q)  r  p  (q  r)?
 Are the statements (p  q)  r  p  (q  r) logically
equivalent?

13. TAUTOLOGY:
 A tautology is a statement form that is always true
regardless of the truth values of the statement variables.
 A tautology is represented by the symbol “T”..
 EXAMPLE:
The statement form p  ~ p is tautology
p  ~p  t
p ~ p p  ~ p
T F T
F T T

14. CONTRADICTION:
 A contradiction is a statement form that is always false
regardless of the truth values of the statement variables.
 A contradiction is represented by the symbol “c”.
 Note:
So if we have to prove that a given statement form is
CONTRADICTION we will make the truth table for
the statement form and if in the column of the given
statement form all the entries are F ,then we say that
statement form is contradiction.

15. EXAMPLE:
 The statement form p  ~ p is a contradiction.
 Since in the last column in the truth table we have F in all
the entries so is a contradiction
p  ~p c
p ~ p p  ~
p
T F F
F T F

16. REMARKS:
 Most statements are neither tautologies nor contradictions.
 The negation of a tautology is a contradiction and vice versa.

17. LOGICAL EQUIVALENCE INVOLVING
TAUTOLOGY
 Show that p  t  p
 Since in the above table the entries in the first and last
columns are identical so we have the corresponding
statement forms are Logically Equivalent that is
p  t  p
p t p  t
T T T
F T F

18. LOGICAL EQUIVALENCE INVOLVING
CONTRADICTION
 Show that p  c  c
 Same truth values in the indicated columns so pc  c
p c p  c
T F F
F F F

19. EXERCISE:
 Use truth table to show that (p  q)  (~p  (p 
~q)) is a tautology.
 Use truth table to show that (p  ~q)  (~p  q) is a
contradiction.

20. INEQUALITIES AND DEMORGAN’S LAWS
 Use De-Morgan’s Laws to write the negation of
-1 < x  4
-1 < x  4 means x > –1 and x  4
By DeMorgan’s Law, the negation is:
Which is equivalent to: x  –1 or x > 4

21. LAWS OF LOGIC
 Given any statement variables p, q and r, a tautology t and
a contradiction c, the following logical equivalences hold:
 Commutative Laws: p  q  q  p
p  q  q  p
 Associative Laws: (p  q)  r  p  (q
 r)
(p  q)  r  p  (q
 r)

22.  Distributive Law: p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p
 r)
 Identity Laws: p  t  p
p  c  p
 Negation Laws: p  ~ p  t
p  ~ p  c

23.  Double Negation Law: ~ (~p)  p
 Idempotent Laws: p  p  p
p  p  p
 De Morgan’s Laws: ~(p  q)  ~p  ~q
~(p  q)  ~p  ~q

24.  Universal Bound Law: p  t  t
p  c  c
 Absorptions Laws: p  (p  q)  p
p  (p  q)  p
 Negation of t and c: ~ t  c
~ c  t

25. Absorptions Laws

26. Absorptions Laws