1. 1 2.2 Definitions & Biconditional Statements Objective: To write biconditionals and recognize good definitions.

2. 2 Biconditional Definition When a conditional and its converse are both true, you can combine them A Biconditional statement combines the conditional and its converse with the word AND In math, biconditionals are written using IF AND ONLY IF

3. 3 Example 1: Biconditional Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional: If the battery-powered calculator runs, then the batteries are good. Converse: If the batteries are good, then the battery-powered calculator runs. Biconditional: The battery-powered calculator runs if and only if the batteries are good.

4. 4 Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional: If Jen is a member, then she has paid her $5 fee. Converse: If Jen has paid her $5 fee, then she is a member. Biconditional: Jen is a member if and only if she has paid her $5 fee. Example 2: Biconditional

5. 5 Ex 3: Separating a Biconditional into Parts Write two statements that form this biconditional. Biconditional: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Conditional: If a number is divisible by 3, then the sum of its digits is divisible by 3. Converse: If the sum of a number’s digits is divisible by 3, then the number is divisible by 3.

6. 6 Write two statements that form this biconditional. Biconditional: A number is prime if and only if it only has two distinct factors, 1 and itself. Conditional: If a number is prime, then it only has two distinct factors, 1 and itself. Converse: If a number has two distinct factors, 1 and itself, then it is prime. Ex 4: Separating a Biconditional into Parts

7. 7 Characteristics of Good Definitions Uses clearly understood terms Is precise (Avoid sort of, almost, etc.) Is reversible (Can be rewritten as a biconditional)

8. 8 Ex 5: Writing a Definition as a Biconditional Definition: Perpendicular lines are two lines that intersect to form right angles. Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

9. 9 Examples of Bad Definitions Definition: An airplane is a vehicle that flies. –Not reversible: If a vehicle flies, then it is an airplane. (What about helicopters?) Definition: A triangle has sharp corners. –Not reversible: If a shape has sharp corners, then it is a triangle. (What about a rhombus with sharp corners?)

10. 10 Biconditional Statements A biconditional combines p → q and q → p as p ↔ q.

11. 11 Assignment Page 78-79 #1-23; 36-46