2 2 Biconditionals Definitions Essential Questions 1 How

2. 2 Biconditionals & Definitions Essential Questions: 1) How do you write biconditionals? 2) What is a good definition?

• When a conditional and its converse are true, you can combine them as a true biconditional, using “and” or “if and only if”. • This is the statement you get by connecting the conditional and its converse with the word and. • You can write a biconditional more concisely, however, by joining the two parts of each conditional with the phrase if and only if.

Practice • Consider this true conditional statement. – What is its converse? • If the converse is also true, combine the statements as a biconditional. – If three points are collinear, then they lie on the same line.

• You can write a biconditional as two conditionals that are converses of each other. • Example: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. – If a number is divisible by 3, then the sum of its digits is divisible by 3. – If the sum of a number’s digits is divisible by 3, then the number is divisible by 3.

Practice • Write two statements that form this biconditional about integers greater than 1: – A number is prime if and only if it has only two distinct factors, 1 and itself. – This one is about lines: Lines are skew if and only if they are noncoplanar.

• A biconditional combines p q and q p as p< -->q. ? ? ?

• A good definition is a statement that can help you identify or classify an object. • A good definition has several important components. – A good definition uses clear understood terms. The terms should be commonly understood or already defined. – A good definition is precise. Good definitions avoid words such as large, sort of, and almost. – A good definition is reversible. That means that you can write a good definition as a true biconditional.

• Example: Writing a Definition as a Biconditional. – Show that the definition of perpendicular lines is reversible. Then write it as a true biconditional. – Definition: Perpendicular lines are two lines that intersect to form right angles. – Conditional: If two lines are perpendicular, then they intersect to form right angles. – Converse: If two lines intersect to form right angles, then they are perpendicular. • The two conditionals, converses of each other, are true, so the definition can be written as a true biconditional. – Biconditional: Two lines are perpendicular if and only if they intersect to from right angles.

Practice • Show that this definition of right angle is reversible. Then write it as a true biconditional. – Def: A right angle is an angle whose measure is 90 degrees. – Cond: If an angle is a right angle, then its measure is 90 degrees. – Conv: If an angle has measure of 90 degrees, then it is a right angle. • The two statements are true. – Bicond: An angle is a right angle if and only if its measure is 90 degrees.

Practice • Determine if the statement is a good definition. Explain. – An airplane is a vehicle that flies. – A triangle has sharp corners. – A square is a figure with four right angles.

Summary • Answer the essential questions in complete sentences. 1) How do you write biconditionals? 2) What is a good definition? • STUDY QUESTIONS: Write 1 -3 study questions in the left column to help explain the notes.