PROPOSITION AND NOT PROPOSITION
Proposition Proposition is a sentence which can explain something true or false. A proposition is usually symbolized by small letter, such as a, b, c, etc.
Examples: a : 2 is an even number (true) b : 4 can be divided by 3 (false) The truth proposition has truth value T (true), while the false proposition has the truth value F (false). The truth value of proposition can be denoted as Ď„ (tau) Examples: a : 8 is an even number, it is a truth proposition, Ď„(a) = T p : 5 is less than 4, it is a false proposition, Ď„(p) = F Hal.: 2
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Open Sentences (Not Proposition) Open sentences containing variables, so can not be determined the truth value yet. Examples : 1. 2 x + 11 = 8 2. It is liquid A. NEGATION If p is a proposition, then the negation of p is written by ~p
and it is read as not p or p is not true. Examples : p : 7 is prime number, then ~p: 7 is not prime number q : 3+2 is equal to 6, then ~q: 3+2 is not equal to 6 The truth table p
~ p
T F
F T
Hal.: 3
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Disjunction Disjunction is two proposition which uses a connective “OR” Proposition disjunction of p and q, denoted as
p∨q
read as p or q
The truth table of disjunction is as follows: p
q
p∨q
T T S S
T F T F
T T T F
Hal.: 4
Sentences to remember : “ students, you have to bring pencil or pen ”
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Conjunction Conjunction is a kind of compound proposition, which uses a connective “AND” Conjunction of proposition p and q denoted as
The truth table of p ∧ q p B B S S
Hal.: 5
q
p∧q
B S B S
B S S S
p∧q
and read p dan q
Sentences to remember : “ students, you have to bring book and pen ”
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Implication Implication is a compound proposition which is formed from two proposition of p and q denoted as if p then q The implication if p then q is denoted as
Read if p then q or • p only if q • q if p • p is sufficient condition for q • q is necessary condition for p
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p⇒q The truth table of implication is as follows:
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p
q
p⇒q
T T F F
T F T F
T F T T
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Biimplication Biimplication from propositions p and q can be denoted as
p⇔q
and read as • • • •
p if and only if q If p then q and if q then p p is necessary condition and enough for q q is necessary condition and enough for p The truth table :
Hal.: 7
p
q
p⇔q
T T F F
T F T F
T F F T
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Compound Proposition Compound proposition is a proposition formed from a few single sentences. They are AND (Λ), OR (V), IF …, THEN …( =>), AND IF ONLY IF … () The examples compound proposition 1. ~ p ∧ q 2. ( p ∨ ~ q ) ⇒ p Examples : Determine the truth value of
( p∨ ~ q) ⇒ p
To determine the truth value, usually using the truth table : p
q
~q
(pv~q)
(pv~q)=>p
T
T
F
T
T
T
F
T
T
T
F
T
F
F
T
F
F
T
T
F
Hal.: 8
So, the truth value of (pv~q) => p is T,T,T,F or denoted as :
τ [( p∨ ~ q ) ⇒ p] =
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TTTF
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Tautology Tautology is a compound proposition which always true for all possible valuation of proposition component Example : Show that the compound proposition Table
p
q
(pvq)
T T F F
T F T F
T T T F
p ⇒ ( p ∨ q) adalah sebuah tautologi p => (pvq) T T T T
So the proposition p ⇒ ( p ∨ q ) is Tautology Kontradiction Kontradiction is a compound proposition which always false for all possible valuation of proposition component
Hal.: 9
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The Equivalence of Two Compound Proposition The two compound proposition is called equivalent if those two compound sentences have the same truth value for all possible valuation of proposition component The equivalence of two compound proposition is denoted by p ≡ q Equivalent
~ ( p ∨ q) ≡ (~ p ∧ ~ q ) ~ ( p ∧ q ) ≡ (~ p ∨ ~ q )
~ ( p ⇒ q) ≡ ( p ∧ ~ q)
( p ⇒ q ) ≡ (~ p ∨ q ) ~ ( p ⇔ q) ≡ ( p ∧ ~ q ) ∨ (q ∧ ~ p ) Hal.: 10
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The Equivalence of Two Compound Proposition ~ ( p ∨ q ) ≡ (~ p ∧ ~ q ) p : Mother takes my sister , q : I am studying (p V q) : Mother takes my sister or, I am studying ~(p V q) : (~p∧~q) = Mother doesn’t take my sister and I am not studying ~ ( p ⇒ q) ≡ ( p ∧ ~ q)
p : I pass to the next grade , q : I get present p⇒q : If I pass the next grade then I get present ~(p⇒q) =(p∧~q) : I pass to the next grade and I don’t get present I pass to the next grade but I don’t get present
Hal.: 11
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The Equivalence of Two Compound Proposition In disjunction and conjuction, they have commulative law, associative law, and distributive law. Commutative Law
Associative Law
p∨q ≡q∨ p
( p ∨ q ) ∨ r ≡ p ∨ (q ∨ r )
p∧q ≡ q∧ p
( p ∧ q ) ∧ r ≡ p ∧ (q ∧ r )
Distributive Law Distributive conjunction to disjunction
p ∨ (q ∧ r ) ≡ ( p ∨ q) ∧ ( p ∨ r ) Distributive conjunction to disjunction
p ∧ (q ∨ r ) ≡ ( p ∧ q) ∨ ( p ∧ r ) Hal.: 12
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The Correlation Between Converse, Inverse, Controposition and Implication If we have an implication p ⇒ q, then we can mate another implication, they are :
q⇒ p
, is called converse from implication p ⇒ q
~ p ⇒~ q
, is called inverse from implication p ⇒ q
~ q ⇒ ~ p , is called contraposition from implication p ⇒ q p
q
~p
~q
B
B
S
S
B
B
B
B
B
S
S
B
S
S
B
B
S
B
B
S
B
B
S
S
S
S
B
B
B
B
B
B
p⇒q
~q⇒~p
q⇒p
~p⇒~q
p ⇒ q ≡ ~ q ⇒~ p
Implication equivalent with its contraposition
q ⇒ p ≡ ~ p ⇒~ q
Converse equivalent with its inverse
Hal.: 13
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Universal Quantor All students of grade X of senior high school 1 are clever The words “All” every are universal equator (general) The symbol of universal equator are : ∀x, p( x) Read as for all x, we have p(x) or ∀x ∈ S , p ( x)
Read as, for all x is S member, we have p(x)
Exsistential Quantor A few students of grade X of senior high school are clever. The word “a few” or “there are/there is” are exsistential quantor (specific) Examples: U = a set of all students of senior high school in Jakarta A = a set of all students of senior high school 1 B = a set of all grade X students of senior high school 1 who are clever The proposition of “a few of grade X students of senior high school ! are clever”, can be denoted by ∃x, x ∈ A dan x ∈ B Read : a few students of senior high school 1 are clever, OR at least a student of grade X of senior high school 1 is clever Hal.: 14
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NEGATION OF QUANTORED STATEMENT Negation of quantored statement
~ [∀x, p ( x)] ≡ ∃x, ~ p ( x) ~ [∃x, p( x)] ≡ ∀x, ~ p ( x)
Example: p : All the first grade students are diligent ~p : There is one students who is not diligent q : There is one students whose house is in Kelapa Gading ~q : All of the first grade students’ house are not in Kelapa Gading r : If all the first grade students pass to the next grade, then I am happy ~r : All the first grade students pass to the next grade and I am not happy ~r : All the first grade students pass to the next grade but I am not happy Hal.: 15
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Drawing Conclusion Drawing Conclusion The statements which have the truth value is called premise Then, using logical principle it can be drawn a new statement ( conclusion) Drawing conclusion is also called argumentation An argumentation is called legal if the premises are true, then the conclusion are also true
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Drawing conclusion Continuation
1. SYLLOGISM
p⇒q
premise 1
q⇒r
premise 2
p⇒r
conclusion
Example: If today is raining, then I will not go to school If I’m not going to school, then my father will angry
premise 1 premise 2
Then, the conclusion is : if today is raining, then my father will angry Hal.: 17
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Drawing Conclusion 2. Ponen Modus p⇒q
premise 1
p
premise 2
q
conclusion
Example: If I have a lot of money, then I will buy a house I have a lot of money
premise 1 premise 2
Then the conclusion is I will buy a house
Hal.: 18
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Drawing Conclusion 3. Tollens Modus p⇒q
premise 1
~q
premise 2
~p
conclusion
Example: If the day the weather is fine, I will come to your party I won’t come to your party
premise 1 premise 2
Then the conclusion is Today the weather is not fine An argumentation is called legal if the conjunction of the premises implicate with the conclusion. And it is called TAUTOLOGY Hal.: 19
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No NoLazy Lazy student! student!
Or Or STUDY STUDYIN INTHE THE WHOLE WHOLELIFE LIFE
Hal.: 20
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