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The Foundation: LogicPropositional Logic, Propositional Equivalence
Muhammad Ariefdownload dari http://arief.ismy.web.id
Propositions / Statements
• A statement (or proposition) is a sentence that is true or false but not both.
• The truth value of a proposition is either TRUE / T / 1 or FALSE / F / 0.
• Ex.– two plus two equals four
• Proposition? Yes• Truth value: true
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Examples• Two plus two equals five
– Proposition? Yes– Truth value: False
• An elephant is bigger than an ant– Proposition? Yes– Truth value: true
• He is a university student – Proposition? No– Truth value: depend on who he is
• C is bigger than 10– Proposition? No – Truth value: unknown
• F plus G equals 9– Proposition? No– Truth value: unknown
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Examples
• Dimana letak kampus UMN – Proposition? No (pertanyaan)
• Jangan memakai sandal ke kampus– Proposition? No (perintah)
• Mudah-mudahan jalan tidak macet – Proposition? No (harapan)
• Indahnya bulan purnama– Proposition? No (ketakjuban / keheranan)
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Compound Propositions / Compound Statements
• A composition of two or more proposition / statement that is true or false but not both
• Example:– Budi is studying at UMN, he is a university student
• Compound statement? Yes• Truth value : True
– Jika x = 1 dan y = 2 maka x lebih besar daripada y• Compound Statement? Yes• Truth value: False
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Formalization of (Compound) Statements
• Translating a (compound) statement to symbols (such as x, y, z) and logical operator.
• Logical operator:~, ¬ not
and
or
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Example~p : not p, negation of p
p q : p and q, conjunction of p and q
p q : p or q, disjunction of p and q
• Order of operation : ( … )
~
Example:
~p q = (~p) qp q r is ambiguous, (p q) r or p (q r)
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Example
• p = it is hot; q = it is sunny
• It is not hot but sunny– It is not hot and it is sunny ~p q
• It is neither hot nor sunny– It is not hot and it is not sunny ~p ~q
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Example
• x ≤ a means x < a or x = a
• a ≤ x ≤ b means a ≤ x and x ≤ b
• 2 ≤ x ≤ 1– compound statement? Yes– Truth value: False
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Truth TableThe list of all possible truth values of a
compound statement.
Truth Table for Negation
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Evaluating the Truth of more General Compound Statements
~p q = (~p) q
Steps: - Evaluate the expressions within the
innermost parentheses- Evaluate the expressions within the next
innermost set of parentheses- Until you have the truth values for the
complete expression.http://arief.ismy.web.id
Evaluating the Truth of more General Compound Statements
p q ~p ~p q
T T F F
T F F F
F T T T
F F T Fhttp://arief.ismy.web.id
Logical EquivalenceDefinition:• Two statement forms are called logically
equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variable.
P = p qQ = q p
• The logical equivalence of statement forms P and Q is denoted by writing P Q.
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De Morgan’s LawsDefinition:• The negation of an AND statement is
logically equivalent to the OR statement in which each component is negated.
~(p q) ~p ~q
• The negation of an OR statement is logically equivalent to the AND statement in which each component is negated.
~(p q) ~p ~qhttp://arief.ismy.web.id
Tautologies and Contradictions
• A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables.
p ~p
• A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables.
p ~p
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Examples
• x + y > 0 (not a statement)
• For x = 1 and y = 2, x + y > 0
• For x = -1 and y = 0, x + y > 0
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Applying De Morgan’s Laws
• John is six feet tall and he weighs at least 200 pounds.• The bus was late or Tom’s watch was slow.• -1 < x 4• p: jim is tall and jim is thin
• John is not six feet tall or he weighs less than 200 pounds.
• The bus was not late and Tom’s watch was not slow.• -1 < x and x 4• -1 < x or x 4• -1 x and x > 4• ~p: jim is not tall or jim is not thin
http://arief.ismy.web.id