3.1 Statements, Negations, and Quantified Statements Chapeter 3 Logic 3.1 Statements, Negations, and Quantified Statements

Statements Examples of statements: London is the capital of England. William Shakespeare wrote the last episode of The Sopranos. The area of a triangle is (1/2)(height)(base)

Not Statements Examples: Commands, questions, and opinions, are not statements because they are neither true or false. Examples: Titanic is the greatest movie of all time. (opinion) Read pages 23 – 57.(command) If I start losing my memory, how will I know? (question)

Using Symbols to Represent Statements In symbolic logic, we use lowercase letters such as p, q, r, and s to represent statements. Example: p: London is the capital of England. q: William Shakespeare wrote the last episode of The Sopranos.

Negation of a Statement The negation of a statement has a meaning that is opposite that of the original meaning. The negation of a true statement is a false statement and The negation of a false statement is a true statement

Negation of a Statement Examples: Statement: Hawaii’s population is smaller than that of California. Negation: Hawaii’s population is not smaller than that of California. Statement: Today’s weather is not very good. Negation: Today’s weather is very good.

Your Turn What is the negation of the following statements? Lenovo is an American brand computer. Lenovo is not an American brand computer. Honda is an American brand car. Honda is not an American brand car. Maui is not the largest island in the state of Hawaii. Maui is the largest island in the state of Hawaii.

Using Symbols to Represent Statements Let p and q represent the following statements: p: Shakespeare wrote the last episode of The Sopranos. q: Today is not Monday. Express each of the following statements symbolically: ~p: Shakespeare did not write the last episode of The Sopranos. ~q: Today is Monday.

Quantified Statements Quantifiers: The words all, some, and no (or none). Statements containing a quantifier: All poets are writers. Some people are bigots. No math books have pictures. Some students do not work hard.

Equivalent Ways of Expressing Quantified Statements

Negation of Quantified Statements The statements diagonally opposite each other are negations. Using Venn diagrams.

Negating a Quantified Statement The mechanic told me, “All piston rings were replaced.” I later learned that the mechanic never tells the truth. What can I conclude? Begin with the mechanic’s statement: All piston rings were replaced. Because the mechanic never tells the truth, I can conclude that the truth is the negation of what I was told.

Negating a Quantified Statement The mechanic told me, “All piston rings were replaced.” I later learned that the mechanic never tells the truth. What can I conclude? The negation of “All A are B” is “Some A are not B”. Thus, I can conclude that Some piston rings were not replaced. I can also correctly conclude that: At least one piston ring was not replaced.