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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

~ (~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

~ (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

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

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

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

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