Statements and Quantifiers Lesson 5.2 Statements and Quantifiers pp. 166-168

Objectives: 1. To define statement as used in logic. 2. To apply quantifiers to statements. 3. To form negations of statements and symbolize them.

Definition A statement is a sentence that is either true or false, but not both.

Determine whether each of the following are statements, and if so, whether each is true or false. x + 7 = 4 1. Not a statement 2. True statement 3. False statement

Determine whether each of the following are statements, and if so, whether each is true or false. 3 + 5 = 6 1. Not a statement 2. True statement 3. False statement

Determine whether each of the following are statements, and if so, whether each is true or false. This sentence is false. 1. Not a statement 2. True statement 3. False statement

Determine whether each of the following are statements, and if so, whether each is true or false. 3x - 9 = 3(x - 3) 1. Not a statement 2. True statement 3. False statement

In mathematical logic, statements are usually symbolized by letters such as p or q. p: Dogs have fleas. q: Baseball is fun.

The negation of the statement Geometry is fun The negation of the statement Geometry is fun! (which is a true statement) is Geometry is not fun! (a false statement). In symbolic logic if the first statement is p, then the negation is ~p.

In logical reasoning the words all and every are represented by an upside down A (). All is called the universal quantifier.

To negate a universal quantifier you must just show the statement is not true for all. p: All flowers are pretty. ~p: There exist flowers that are not pretty.

To negate a universal quantifier you must just show the statement is not true for all. p: All flowers are pretty. ~p: Not all flowers are pretty.

To negate a universal quantifier you must just show the statement is not true for all. p: All flowers are pretty. ~p: Some flowers are not pretty.

Another type of quantifier is the existential quantifier (), which implies “one or more.”

To negate the existential quantifier the statement cannot be true for any. p: There exists dogs that have fleas. ~p: No dogs have fleas.

Examples p: Girls are logical. p: ~p: p: ~p: All girls are logical. Not all girls are logical. Some girls are logical. No girls are logical.

Examples p: Cars are expensive. p: ~p: p: ~p: All cars are expensive. Not all cars are expensive. Some cars are expensive. No cars are expensive.

Homework pp. 167-168

►B. Exercises Write the following symbolizations in words given: p: Ben Franklin was an inventor. q: Prime numbers are divisible by 2. r: Men are bald. s: Students should study mathematics. 13.  r

►B. Exercises Write the following symbolizations in words given: p: Ben Franklin was an inventor. q: Prime numbers are divisible by 2. r: Men are bald. s: Students should study mathematics. 15. ~q

►B. Exercises Write the following symbolizations in words given: p: Ben Franklin was an inventor. q: Prime numbers are divisible by 2. r: Men are bald. s: Students should study mathematics. 17.  s

►B. Exercises Write the following symbolizations in words given: p: Ben Franklin was an inventor. q: Prime numbers are divisible by 2. r: Men are bald. s: Students should study mathematics. 19. ~ r

■ Cumulative Review True/False 26. All obtuse triangles are scalene.

■ Cumulative Review True/False 27. A rhombus is a trapezoid.

■ Cumulative Review True/False 28. Some triangles are not convex.

■ Cumulative Review True/False 29. No cones are polyhedra.

■ Cumulative Review True/False 30. Some half-planes intersect.