Discrete Structures

1. PREPOSITIONal
LOGIC

2. A statement is a declaratory sentence which is true or
false but not both. In other words , a statement is a
declarative sentence which has a definate truth table.

3. Logical connectives or sentence
connectives
These are the words or symbols used to combine
two sentence to form a compound statement.
logic Name rank
~ Negation 1
^ Conjunction 2
V Disjunction 3
=> Conditional 4
 Biconditional 5

4. A B ^ V ~A =>  NOR NAND XOR EX-
NOR
T T T T F T T F F F T
T F F T F F F F T T F
F T F T T T F F T T F
F F F F T T T T T F T

5. TAUTOLOGY
i. A TAUTOLOGY IS A PREPOSITION WHICH IS
TRUE FOR ALL TRUTH VALUES OF ITS SUB-
PREPOSITIONS OR COMPONENTS.
ii. A TAUTOLOGY IS ALSO CALLED LOGICALLY
VALID OR LOGICALLY TRUE.
iii. ALL ENTRIES IN THE COLUMN OF
TAUTOLOGY ARE TRUE.

6. For example:
p^q=>q
P q p^q q p^q=>
q
T T T T T
T F F F T
F T F T T
F F F F T

7. Contradiction
 CONTRADICTION IS A PREPOSITION WHICH IS
ALWAYS FALSE FOR ALL TRUTH VALUES OF ITS
SUB-PREPOSITIONS OR COMPONENTS.
 A CONTRADICTION IS ALSO CALLED LOGICALLY
INVALID OR LOGICALLY FALSE
 ALL ENTRIES IN THE COLUMN OF
CONTRADICTION ARE FALSE.

8. FOR EXAMPLE
(P v Q)^(~P)^(~Q)
P Q PVQ ~P ~Q (P v Q)^(~P)^(~Q)
T T T F F F
T F T F T F
F T T T F F
F F F T T F

9. Contingency
It is a preposition which is either true or
false depending on the truth value of its
components or preposition..

10. FOR EXAMPLE
~p ^ ~q
p q ~p ~q ~p ^ ~q
T T F F F
T F F T F
F T T F F
F F T T T

11. Logical equivalence
Two statements are called logically equivalent if the truth
values of both the statements are always identical..
For example:
If we take two statements p=>q and ~q =>~p , then there
truth table values must be equal to satisfy the condition of
logical equivalence..

12. p q ~p ~q p=>q ~q=>~p
T T F F T T
T F F T F F
F T T F T T
F F T T T T
SINCE,THE TRUTH TABLE VALUES OF BOTH
STATEMENTS IS SAME. THUS, THE TWO
STATEMENTS ARE LOGICALLY EQUIVALENT..

13. LOGICAL IMPLICATIONS
 DIRECT IMPLICATION (p=>q)
 CONVERSE IMPLICATION (q=>p)
 INVERSE OR OPPOSITE IMPLICATION (~p=>~q)
 CONTRAPOSITIVE IMPLICATION (~q=>~p)

14. Algebra of
preposition
1) Commutative law
2) Associative law
3) Distributive law
4) De Morgan’s law
5) Idempotent law
6) Identity law

15. Idempotent law
1. pVpp
2. p^pp
p p pvp p v pp p^p p^ pp
T T T T T T
F F F F F F

16. Commutative law
• pvq=qvp
• p^q=q^p
p q pvq qvp p^q q^p
T T T T T T
T F T T F F
F T T T F F
F F F F F F

17. Associative law
• (p v q) v r  p v (q v r)
• (p ^ q) ^ r  p ^ (q ^ r)
p q r pvq ( p v q) v r qVr p v (q v r)
T T T T T T T
T T F T T T T
T F T T T T T
T F F T T F T
F T T T T T T
F T F T T T T
F F T F T T T
F F F F F F F

18. Distributive law
• p ^ (q v r)  (p ^ q) v (p ^ r)
• p ^ (q v r)  (p ^ q) v (p ^ r)
p q r qvr p^(q v r) p^q p^r (p^q)v(p^r)
T T T T T T T T
T T F T T T F T
T F T T T F T T
T F F F F F F F
F T T T F F F F
F T F T F F F F
F F T T F F F F
F F F F F F F F

19. De Morgan’s law
• ~(p v q)  ~p ^ ~q
• ~(p ^ q)  ~p v ~q
p q (p v q) ~(p v q) ~p ~q ~p ^ ~q
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T T T

20. Identity law
1) p ^ T  p 2) T ^ p  p
3) p v F  p 4) F v p  p
P T P^T P F P v F
T T T T F T
F T F F F F

21. TRANSITIVE RULE
pq
qr
--------------
pr
Rule of detachment
P
Pq
----------
q

22. EXAMPLE
TEST THE VALIDITY OF THE FOLLOWING
ARGUMENT….
IF A MAN IS A BACHELOR,HE IS WORRIED(A PREMISE)
IF A MAN IS WORRIED,HE DIES YOUNG(A PREMISE)
-----------------------------------------------------------------------------------------------------
BACHELORS DIE YOUNG(CONCLUSION)
P: A man is a bachelor
Q:he is worried
R: he dies young

23. The given argument in symbolic form can be
written as:
pq (a premise)
qr (a premise)
--------------------
pr (conclusion)
The given argument is true by law of
syllogism(law of transitive)…

24. p q r pq qr pr pq ^ qr (pq) ^ (qr)
=> pr
T T T T T T T T
T T F T F F F T
T F T F T T F T
T F F F T F F T
F T T T T T T T
F T F T F T F T
F F T T T T T T
F F F T T T T T

25. PRESENTATION BY :
ASHWINI VIPAT