1. Citra Noviyasari, S.Si, MT
Propositions
Citra Noviyasari, S.Si, MT
Discrete - Citra N., S.Si, MT

2. Definition
A proposition is a (statements) declarative sentences that is either true or false (but not both).
For instance :
“Paris is in France” (true)
“London is in Denmark” (false)
“2 < 4” (true),
“4 = 7 (false)”.
However the following are not propositions:
“what is your name?”
“do your homework”
“x is an even number”
Discrete - Citra N., S.Si, MT

3. Letters are used to denote propositions, the conventional letters used for this purpose are p, q, r, s.
The truth value of a propositions is true, denoted by T, if it is a false propositions denoted by F.
Many mathematical statements are constructed by combining one or more propositions. New propositions called compound propositions.
Discrete - Citra N., S.Si, MT

4. Truth Tables Connectives are used for making compound propositions.
The main ones are the following (p and q represent given propositions):
Name
Represented
Meaning
Negation ~p
“not p”
Conjunction
p Λ q
“p and q”
Disjunction
p V q
p or q (or both)
Implication
p  q
“if p then q”
Biconditional
p ↔ q
“p if and only if q”
Discrete - Citra N., S.Si, MT

5. How to translate sentences into expressions involving propositions variables and logical connectives
You can acces the Internet from campus only if you are a computer science major or you are not a freshman
p : you can access the internet from campus
q : you are a computer science major
r : You are a freshman
Expressions : p  (q V ~r)
Discrete - Citra N., S.Si, MT

6. p : You can ride the roller coaster q : You under 4 feet tall
You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.
p : You can ride the roller coaster
q : You under 4 feet tall
r : You are older than 16 years old
Discrete - Citra N., S.Si, MT

7. p : Ratingnya berbintang 3 q : Makanannya enak r : Pelayanannya baik
Ratingnya berbintang 3 jika dan hanya jika makanannya enak atau pelayanannya baik, atau keduanya
p : Ratingnya berbintang 3
q : Makanannya enak
r : Pelayanannya baik
Discrete - Citra N., S.Si, MT

8. The truth or falsehood of a proposition is called its truth value
The truth value of a compound proposition depends only on the value of its components.
The truth or falsehood of a proposition is called its truth value p q
~ p
p Λ q
p V q
p  q
p ↔ q T F
Discrete - Citra N., S.Si, MT

9. Example for truth table : ¬(p Λ q)  p
Discrete - Citra N., S.Si, MT

10. Try to make truth table from these propositions
~ (~p Λ q) Λ (p V q)
~ (~p Λ q) ↔ (p  q)
p V (q Λ r)
Discrete - Citra N., S.Si, MT

11. Bit operations
Computers represent infomation using bits (binnary digit).
A bit has two possible values, namely, 0 (zero) and 1 (one). 1 represent true and a 0 bit represent false.
Computer bit operations correspond to the logical connectives x y
x Λ y
x V y 1
Discrete - Citra N., S.Si, MT

12. Logical Equivalences
A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called tautology.
A compound propositions that is always false is called a contradiction
And, a proposition that is neither a tautology nor a contradiction is called a contingency.
Discrete - Citra N., S.Si, MT

13. Examples p V ~(p Λ q) (p Λ q) Λ ~(p V q) p q p Λ q ~(p Λ q)T F
(p Λ q) Λ ~(p V q) p q
p Λ q
p V q
~(p V q)
(p Λ q) Λ ~(p V q) T F
Discrete - Citra N., S.Si, MT

14. Examples (p Λ q) V (~ q Λ r) p Q r p Λ q ~q Λ r (p Λ q) V (~ q Λ r) TF
Discrete - Citra N., S.Si, MT

15. Logical Equivalences
When two compound propositions have the same truth values no matter what truth value their constituent propositions have, they are called logically equivalent. (notation :  )
For instance : p  q and ~p V q are logically equivalent, and we write it :
p  q  ~p V q p q
~ p
p  q
~p V q T F
Discrete - Citra N., S.Si, MT

16. Try to figure out, which one is equivalence!
~ (p V ~q) V ( ~p Λ ~q)  ~p
~ ((~p Λ q) V (~p Λ ~q)) V (p Λ q)  p
p V (p Λ r)  (p V q) Λ (p V r)
Discrete - Citra N., S.Si, MT

17. Properties of Propotions
The propositions operations verify the following properties:
Table Logical Equivalences
Name
Equivalence
Identity Laws
p V S  p
p Λ B  p
2. Bound Laws
p Λ S  S
p V B  B
3. Complement Laws
p V ~p  B
p Λ ~p  S
4. Idempotent Laws
p V p  p
p Λ p  p
Discrete - Citra N., S.Si, MT

18. Table Logical Equivalences Name Equivalence
5. Involution Laws
~(~p)  p
6. Absorption Laws
p V (p Λ q)  p
p Λ (p V q)  p
7. Commutative Laws
p V q  q V p
p Λ q  q Λ p
8. Associative Laws
p V (q V r)  (p V q) V r
p Λ (q Λ r)  (p Λ q) Λ r
9. Distributive Laws
p V (q Λ r)  (p V q) Λ (p V r)
p Λ (q V r)  (p Λ q) V(p Λ r)
10. DeMorgan’s Laws
~(p Λ q)  ~p V ~q
~(p V q)  ~p Λ ~q
Discrete - Citra N., S.Si, MT

19. How to show that an expressions are logically equivalent
We will establish this equivalence by developing a series of logical equivalences, using one of the equivalences in table Logical Equivalences at a time.
Example : (p Λ q) V (p Λ~ q)
Solution :
(p Λ q) V (p Λ~ q)
 p Λ (q V ~q) {Distributive Laws)
 p Λ B {Complement laws}
 p {Bound Laws)
Discrete - Citra N., S.Si, MT

20. ~ (~p Λ q) Λ (p V q)  (~ (~p) V ~q) Λ (p V q) ... {De Morgan’s Laws}
 (p V ~q) Λ (p V q) {Involutions Laws}
 p Λ (~q V q) {distributive Law’}
 p Λ B {Complement laws}
 p {Bound Laws}
Discrete - Citra N., S.Si, MT

21. ~ (p V (~p Λ q))  ~ p Λ ~ (~p Λ q)  ~ p Λ (~ (~p) V ~q)
~ (p V (~p Λ q))  ~p Λ ~ q
~ (p V (~p Λ q))
 ~ p Λ ~ (~p Λ q)
 ~ p Λ (~ (~p) V ~q)
 ~ p Λ (p V ~q)
 (~ p Λ p) V (~p Λ ~q)
 F V (~p Λ ~q)
 ~p Λ ~q
Discrete - Citra N., S.Si, MT

22. Try for these propositions
((p V(p Λ~p) Λ(p V ~(p Λq))
((p V q)V~p) V(p V(p Λ q))
Discrete - Citra N., S.Si, MT

23. Converse, Contrapositive
The converse of a conditional proposition p  q is the proposition q  p. As we have seen, the biconditional proposition is equivalent to the conjunction of a conditional proposition an its converse.
The contrapositive of a conditional proposition p  q is the proposition ~q  ~p. They are logically equivalent
Discrete - Citra N., S.Si, MT

24. Assignment (1) If we knew : p , q = T and r, s = F
Define value from these propositions :
p V (q Λ r)
(p Λ q Λ r ) V ~ ((p V q) Λ (r V s))
(~ (p Λ q) V ~r) V (((~ p Λ q) V~ r) Λ s)
From these propositions below make a truth table :
(p Λ q)  (p  q)
~ (p V q) Λ ~ (s V r)
Discrete - Citra N., S.Si, MT

25. Assignment (2)
Find new propositions by developing a series of logical quivalences :
p V(p Λ q) Λ p V ~(p V ~q)
~p Λ ~((p Λ ~q)V(p Λ q))
Discrete - Citra N., S.Si, MT