Statements and Logical Connectives 3.1 Statements and Logical Connectives

Logic and the English Language Connectives - words such as and, or, if, then Exclusive or - one or the other of the given events can happen, but not both. Inclusive or - one or the other or both of the given events can happen.

Statements and Logical Connectives Statement - A sentence that can be judged either true or false. Labeling a statement true or false is called assigning a truth value to the statement. Simple Statements - A sentence that conveys only one idea. Compound Statements - Sentences that combine two or more ideas and can be assigned a truth value.

Quantifiers Negation of a statement - change a statement to its opposite meaning. The negation of a false statement is always a true statement. The negation of a true statement is always a false statement. Quantifiers - words such as all, none, no, some, etc… Be careful when negating statements that contain quantifiers.

Negation of Quantified Statements Form of statement All are. None are. Some are. Some are not. Form of negation Some are not. Some are. None are. All are. None are. Some are not. All are. Some are.

Example: Write Negations Write the negation of the statement. Some candy bars contain nuts. Solution: Since some means “at least one” this statement is true. The negation is “No candy bars contain nuts,” which is a false statement.

Example: Write Negations continued Write the negation of the statement. All tables are oval. Solution: This is a false statement since some tables are round, rectangular, or other shapes. The negation would be “Some tables are not oval.”

Compound Statements Statements consisting of two or more simple statements are called compound statements. The connectives often used to join two simple statements are and, or, if…then…, and if and only if.

Not Statements The symbol used in logic to show the negation of a statement is ~. It is read “not”.

And Statements is the symbol for a conjunction and is read “and.” The other words that may be used to express a conjunction are: but, however, and nevertheless.

Example: Write a Conjunction Write the conjunction in symbolic form. The dog is gray, but the dog is not old. Solution: Let p and q represent the simple statements. p: The dog is gray. q: The dog is old. In symbol form, the compound statement is p ^ ~ q

Or Statements: The disjunction is symbolized by and read “or.” In this book the “or” will be the inclusive or, (except where indicated in the exercise set).

Example: Write a Disjunction Write the statement in symbolic form. Carl will not go to the movies or Carl will not go to the baseball game. Solution: Let p and q represent the simple statements. p: Carl will go to the movies. q: Carl will go to a baseball game. In symbol form, the compound statement is

If-Then Statements The conditional is symbolized by and is read “if-then.” The antecedent is the part of the statement that comes before the arrow. The consequent is the part that follows the arrow.

Example: Write a Conditional Statement Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. a. If Nathan goes to the park, then he will swing. b. If Nathan does not go to the park, then he will not swing. Solutionsa) b)

If and Only If Statements The biconditional is symbolized by and is read “if and only if.” If and only if is sometimes abbreviated as “iff.”

Example: Write a Statement Using the Biconditional Let p: The dryer is running. q: There are clothes in the dryer. Write the following symbolic statements in words. a) b) Solutions: a. The clothes are in the dryer if and only if the dryer is running. b. It is false that the dryer is running if and only if the clothes are not in the dryer.

Homework P. 101 # 1 – 8, 10-82 evens