A Statement; Logical Operations

A Statement; Logical Operations

Mathematical logic is a branch of mathematics that deals with the formal principles, methods and criteria of validity of reasoning and knowledge. Logic is concerned with what is true and how we can know whether something is true. Augustus De Morgan George Boole Bertrand Russel

Main object of mathematical logic is a statement. Definition: A STATEMENT (A PROPOSITION) is a declarative sentence that is true or false, but not both. a statement = ISKAZ

Example: Which of these sentences are statements? Are they true or false? 1) 1 < 2 2) What is your name? 3) 25 = 5 4) x 2 = 4 simple propositions

NOTATION: Often a statement is abbreviated by lower case letter: p,q,r, - propositional variables ( iskazna slova ) Example: p: Seven is a prime number

TRUTH VALUE OF A STATEMENT: If the statement is true, its truth value is T. If the statement is false, its truth value is F. truth value = istinitosna vrednost p T, ako je p tačan iskaz, ako je p netačan iskaz

Like in language, we can build compound sentences using connectives. COMPOUND STATEMENTS are statements with connectives. connective operator notation NOT Negation AND Conjunction OR Disjunction IF THEN Implication IF AND ONLY IF Equivalence

Negation Let p stands for a given statement. The negation of the statement p is the statement NOT p that respresents the logical opposite of p. When p is true, then not p is false and vice versa. ~p p, p NOT p p T F p F T TRUTH TABLE = is a summary of all possible truth values of a logic statement

truth table = tablica istinitosti Example: p: π < 4 true p: π 4 false

Conjunction A conjuction is a compound statement formed by combining two simple statements using the word AND. p, q - statements ( conjuncts ) p q p AND q The conjuction, p and q, is true only when both p and q are true, and false is otherwise.

p q p q T T T T F F F T F F F F n - number of statements 2 n - number of rows in truth tables Example : p: " 2 > 0 " true q: " 2 < 0 " false p q " 2 > 0 and 2 < 0 false

Disjunction A disjuction is a compound statement formed by combining two simple statements using the word OR. p, q - statements ( disjuncts ) p q p OR q The disjuction, p or q, is true when at least one of the statements p and q is true, and is false when both are false.

p q p q T T T T F T F T T F F F Example : p: " 2 > 0 " true q: " 2 < 0 " false p q " 2 > 0 or 2 < 0 true

Implication ( Conditional ) A conditional is a compound statement formed by combining two simple statements using words IF THEN. p, q - statements p q ( p q ) if p then q The implication, if p then q, is false when p is true and q is false. In all other cases, the implication is true.

p q p q p - antecedent, premise, T T T hypothesis, T F F protasis F T T q - consequent, F F T conclusion, apodosis

p q if p, then q p implies q p only if q q if p q provided that p q whenever p q is a necessary condition for p p is a sufficient condition for q Example: The following implication is true: " 3 + 1 = 7 6 1 = 2" If 3 + 1 = 7, then 1 = 7 3 = 4, so 6 1 = 6 4 = 2.

Equivalence ( Biconditional ) A biconditional is the conjuction of the two implications p q and q p. p, q - statements p q p q p if and only if q p iff q if p then q, and if q then p The biconditional, p if and only if q, is true when p and q are both true or both false.

p q p q T T T T F F F T F F F T Example: A number is divisible by 6 if and only if is divisible by 2 and 3.

Every definition is a true biconditional! Example: Line segments that have the same measure are congruent. Restated: Two line segments are congruent if and only if they have the same measure. Biconditionals are used to solve equations! 3x + 7 = 19 if and only if 3x = 12 if and only if x = 4