6.1 Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19 th and early 20 th century. George Boole (Boolean algebra) introduced mathematical methods to logic in 1847 while Georg Cantor did theoretical work on sets and discovered that there are many different sizes of infinite sets.

Statements or Propositions A proposition or statement is a declaration which is either true or false. A proposition or statement is a declaration which is either true or false. Some examples: Some examples: 2+2 = 5 is a statement because it is a false declaration. 2+2 = 5 is a statement because it is a false declaration. Orange juice contains vitamin C is a statement that is true. Orange juice contains vitamin C is a statement that is true. Open the door. This is not considered a statement since we cannot assign a true or false value to this sentence. It is a command, but not a statement or proposition. Open the door. This is not considered a statement since we cannot assign a true or false value to this sentence. It is a command, but not a statement or proposition.

Negation The negation of a statement, p, is “not p” and is denoted by ┐ p The negation of a statement, p, is “not p” and is denoted by ┐ p Truth table: Truth table: p ┐ p p ┐ p TF TF FT FT If p is true, then its negation is false. If p is false, then its negation is true. If p is true, then its negation is false. If p is false, then its negation is true.

Disjunction A disjunction is of the form p V q and is read p or q. A disjunction is of the form p V q and is read p or q. Truth table for disjunction: Truth table for disjunction: pqp V q pqp V q TTT TTT TFT TFT FTT FTT FFF FFF A disjunction is true in all cases except when both p and q are false. A disjunction is true in all cases except when both p and q are false.

Conjunction A conjunction is only true when both p and q are true. Otherwise, a conjunction of two statements will be false: A conjunction is only true when both p and q are true. Otherwise, a conjunction of two statements will be false: Truth table: Truth table: pqp q pqp q TT T TT T TF F TF F FT F FT F FF F FF F

Conditional statement To understand the logic behind the truth table for the conditional statement, consider the following statement. To understand the logic behind the truth table for the conditional statement, consider the following statement. “If you get an A in the class, I will give you five bucks.” “If you get an A in the class, I will give you five bucks.” Let p = statement “ You get an A in the class” Let p = statement “ You get an A in the class” Let q = statement “ I will give you five bucks.” Let q = statement “ I will give you five bucks.” Now, if p is true (you got an A) and I give you the five bucks, the truth value of Now, if p is true (you got an A) and I give you the five bucks, the truth value of p q is true. The contract was satisfied and both parties fulfilled the agreement. p q is true. The contract was satisfied and both parties fulfilled the agreement. Now, suppose p is true (you got the A) and q is false (you did not get the five bucks). You fulfilled your part of the bargain, but weren’t rewarded with the five bucks. Now, suppose p is true (you got the A) and q is false (you did not get the five bucks). You fulfilled your part of the bargain, but weren’t rewarded with the five bucks. So p q is false since the contract was broken by the other party. So p q is false since the contract was broken by the other party. Now, suppose p is false. You did not get an A but received five bucks anyway. (q is true) No contract was broken. There was no obligation to receive 5 bucks, so truth value of p q cannot be false, so it must be true. Now, suppose p is false. You did not get an A but received five bucks anyway. (q is true) No contract was broken. There was no obligation to receive 5 bucks, so truth value of p q cannot be false, so it must be true. Finally, if both p and q are false, the contract was not broken. You did not receive the A and you did not receive the 5 bucks. So p q is true in this case. Finally, if both p and q are false, the contract was not broken. You did not receive the A and you did not receive the 5 bucks. So p q is true in this case.

Truth table for conditional pqp q pqp q TTT TFF FTT FFT

Variations of the conditional Converse: The converse of p q is q p Converse: The converse of p q is q p Contrapositive: The contrapositive of p q is Contrapositive: The contrapositive of p q is ┐q ┐p ┐q ┐p

Examples Let p = you receive 90% Let p = you receive 90% Let q = you receive an A in the course Let q = you receive an A in the course p q ? p q ? If you receive 90%, then you will receive an A in the course. If you receive 90%, then you will receive an A in the course. Converse: q p Converse: q p If you receive an A in the course, then you receive 90% If you receive an A in the course, then you receive 90% Is the statement true? No. What about the student who receives a score greater than 90? That student receives an A but did not achieve a score of exactly 90%. Is the statement true? No. What about the student who receives a score greater than 90? That student receives an A but did not achieve a score of exactly 90%.

Example 2 State the contrapositive in an English sentence: State the contrapositive in an English sentence: Let p = you receive 90% Let p = you receive 90% Let q = you receive an A in the course Let q = you receive an A in the course p q ? p q ? If you receive 90%, then you will receive an A in the course If you receive 90%, then you will receive an A in the course ┐ q ┐ p ┐ q ┐ p If you don’t receive an A in the course, then you didn’t receive 90%. If you don’t receive an A in the course, then you didn’t receive 90%. The contrapositive is true not only for these particular statements but for all statements, p and q. The contrapositive is true not only for these particular statements but for all statements, p and q.

Logical equivalent statements Show that is logically equivalent to Show that is logically equivalent to We will construct the truth tables for both sides and determine that the truth values for each statement are identical. We will construct the truth tables for both sides and determine that the truth values for each statement are identical. The next slide shows that both statements are logically equivalent. The red columns are identical indicating the final truth values of each statement. The next slide shows that both statements are logically equivalent. The red columns are identical indicating the final truth values of each statement.