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The Foundation: Logic Propositional Logic, Propositional Equivalence Muhammad Arief download dari http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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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) http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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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) q p q r is ambiguous, (p q) r or p (q r) http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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Truth Table The list of all possible truth values of a compound statement. Truth Table for Negation http://arief.ismy.web.id

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Truth Table for Conjunction p q http://arief.ismy.web.id

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Truth Table for Disjunction p q http://arief.ismy.web.id

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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

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Evaluating the Truth of more General Compound Statements pq~p ~p q TTFF TFFF FTTT FFTF http://arief.ismy.web.id

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Truth Table for Exclusive Or Definition: (p q) ~(p q) : p q, p XOR q, http://arief.ismy.web.id

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Truth Table for (p q) ~r http://arief.ismy.web.id

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Logical Equivalence Definition: 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 q Q = q p The logical equivalence of statement forms P and Q is denoted by writing P Q. http://arief.ismy.web.id

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Logical Equivalence P Q http://arief.ismy.web.id

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~(~p) p http://arief.ismy.web.id

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Are ~(p q) and ~p ~q logically equivalent? http://arief.ismy.web.id

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De Morgan’s Laws Definition: 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 ~q http://arief.ismy.web.id

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De Morgan’s Laws http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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Tautologies and Contradictions http://arief.ismy.web.id

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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 http://arief.ismy.web.id

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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

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