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Biconditional Statements

Biconditional Statements. Define and write biconditionals State the characteristics of good definitions. 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

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Biconditional Statements

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  1. Biconditional Statements • Define and write biconditionals • State the characteristics of good definitions

  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 • Conditional: If Jen is a member, she has paid her $5 fee. • Converse: If Jen has paid her $5 fee, she is a member. • Biconditional: Jen is a member if and only if she has paid her $5 fee.

  3. Biconditional Example • 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. Separating a Biconditional into Parts • 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.

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

  6. 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.

  7. 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?)

  8. Learning Check and Summary • What two things does a biconditional combine? • What phrase is commonly used in biconditional statements in math? • From a math perspective, why would “A hummingbird is a bird.” be a bad definition?

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