## Presentation on theme: "The Foundation: Logic Propositional Logic, Propositional Equivalence Muhammad Arief download dari"— Presentation transcript:

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

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

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

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

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

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

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

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

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

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

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

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 pq~p ~p  q TTFF TFFF FTTT FFTF http://arief.ismy.web.id

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

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

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

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

~(~p)  p http://arief.ismy.web.id

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

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

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