Section 1-4 Logic Katelyn Donovan MAT 202 Dr. Marinas January 19, 2006

What is a Statement? A statement is a sentence that is either true or false, but not both. Which of the following are statements? 1.Lauren has blue eyes. 2.Bush is the best president. 3.He smells.

How did you do? #1 is a statement since Lauren was identified as the person with blue eyes. #2 is not a statement because it can be true and false, depending on who you ask. #3 is not a statement because the he who smells is not identified.

What is a Negation?  The negation of a statement is a statement with the opposite truth value of the given statement.  If a statement is true, the negation is false and if a statement is false, the negation is true.  Ex. Statement : It is raining now. Negation: It is not raining now.

Truth Tables Truth tables are used to show all possible True-False patterns for statements. The symbol p represents a statement and the symbol ~ p (read as “not p”) represents a negation. The Truth Table for p and ~ p is shown below. Statement p Negation ~ p TF FT

Compound Statements Compound Statements are two statements with a connector such as and/or. The symbol ^ represents and. The symbol v represents or.

Conjunction A conjunction is a compound statement formed by joining two statements with the connector AND. The conjunction "p and q" is symbolized by p ^ q. A conjunction is true when both of its combined parts are true; otherwise it is false. The Truth Table for Conjunction is shown below: pqp ^ q TTT TFF FTF FFF

Disjunction A disjunction is a compound statement formed by joining two statements with the connector OR. The disjunction “p or q” is symbolized by p V q. The disjunction is false only when both p and q are false, everywhere else its true. The Truth Table for Disjunction is shown below: pqp v q TTT TFT FTT FFF

Conditionals Conditionals are statements written as “ if p, then q ” or p  q Conditionals are also known as implications The statement after the “if” is the hypothesis and the statement after the “then” is the conclusion. The Truth Table for conditional (implication) is below: pqp  q TTT TFF FTT FFT

Example of a Conditional IF Ariel kisses Eric, THEN she will remain human. p q Conditional p q Conditional (1)T T Ariel kisses Eric; she will remain T human. (2)T F Ariel kisses Eric; she does not remain F human. (3)F T Ariel does not kiss Eric; she remains human. T (4)F F Ariel does not kiss Eric; she does not remain T human.

Conditionals are written in many ways: If p, then q If p, q q, if p p implies q p only if q p is sufficient condition for q q is a necessary condition for p

Any Implication p  q has three related implication statements: Statement: If p, then q. p  q Converse: If q, then p. q  p Inverse: If not p, then not q. ~p  ~q Contrapositive: If not q, then not p. ~q  ~p

Example Statement: If I am in Miami, then I am in Florida. (p  q) Converse: If I am in Florida, then I am in Miami. (q  p) Inverse: If I am not in Miami, then I am not in Florida. (~p  ~q) Contrapositive: If I am not in Florida, then I am not in Miami. (~q  ~p)

PROPERTY TIME!  Equivalence of a statement and its contrapositive.  The implication p  q and its contrapositive ~q  ~ p are logically equivalent.

Biconditional Statements  Connecting a statement and its converse with the connective and gives (p  q) ^ (q  p )  This compound statement can be written as p   q and is read as “p if and only if q”  This statement is called a biconditional. pq p   q TTT TFF FTF FFT

HAVE FUN AND GOOD LUCK! DO NOT FORGET TO COME SEE ME IN THE MATH LAB FOR ADDITIONAL ASSISTANCE