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HW 1.4(g) Handout # 45-50

HW 1.4(f) Solutions 2. (a) If I follow, then you lead (b) If you do not lead, then I will not follow (c) If I do not follow then you will not lead 4. (a) If I was rich then I would have a nickel for each time that happened. (b) If I did not have a nickel for each time that happened then I would not be rich. (c) If I was not rich then I would not have a nickel for every time that happened. 6. (a) If it contains calcium then it is milk. (b) If it is not milk then it does not contain calcium. (c) If it does not contain calcium then it is not milk

HW 1.4(f) Solutions 8. (a) If it gathers no moss then it is a rolling stone. (b) If it is not a rolling stone then it gathers moss. (c) If it gathers moss then it is not a rolling stone. 10.(a) If there is a fire then there is smoke. (b) If there is no smoke then there is no fire. (c) If there is no fire then there is no smoke.

HW 1.4(f) Solutions 12.(a) (b) (c) 14.(a)

Guiding question: What is a biconditional?

Variations of a Conditional What are the variations of a conditional? We will learn one more related form of a conditional.

The Conditional and Converse Write the converse of the conditional, and determine the truth value. If a substance is water, then its chemical formula is H2O. Converse: If a substance’s chemical formula is H2O, then the substance is water. Truth values: Conditional = True Converse = Note: When a conditional and its converse are true, they can be combined to form a biconditional.

Biconditionals Notice that the biconditional states ‘if p then q and if q then p’. In other words the biconditional states ‘p if and only if q’.

Biconditionals Notice that the biconditional states ‘if p then q and if q then p’. In other words the biconditional states ‘p if and only if q’. Which is represented by the symbol:

Writing a Biconditional Statement Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. If x = 5, then x + 15 = 20. Converse: If x + 15 = 20, then x = 5. TRUE Biconditional: x = 5 if and only if x + 15 = 20. OR x + 15 = 20 if and only if x = 5.

Writing a Biconditional Statement Write the converse of the following true conditional. If the converse is also true, then combine the statements as a biconditional. If three points are collinear, then they lie on the same line.

Separating a Biconditional Statement You can write a biconditional as two conditionals that are the converse of each other. Write the two statements that form this biconditional: Lines are skew if and only if they are noncoplanar. If lines are skew then they are noncoplanar. If lines are noncoplanar, then they are skew.

Separating a Biconditional Statement 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.

Biconditionals and the Real-World In Geometry, and the real-world, definitions are important. A good definition is a statement that can help you to identify or classify an object. A good definition… Is clear and uses terms that are already defined, Is precise and non-ambiguous Is reversible… a true biconditional!

Biconditionals and Definitions Show that this definition of a triangle is reversible. Then write it as a true biconditional. A triangle is a polygon with exactly three sides. Statement 1: If a polygon is a triangle then it has exactly three sides. True Statement 2: If a polygon has exactly three sides then it is a triangle. Biconditional: A polygon is a triangle if and only if it has exactly three sides.

Biconditionals and Definitions Show that this definition of a right angle is reversible. Then write it as a true biconditional. Definition A right angle is an angle whose measure is 90.

Biconditionals and Definitions One way to show that a statement is not a good definition is to find a counterexample. Is the given statement a good definition? Explain. An airplane is a vehicle that flies.  Statement 1: If a vehicle is an airplane, then it flies. TRUE Statement 2: If a vehicle flies, then it is an airplane. FALSE Counterexample: A helicopter is a vehicle that flies, but a helicopter is not an airplane. The statement is not a good definition because it is not reversible.

Biconditionals and Definitions Is the given statement a good definition? Explain. A triangle has sharp corners.  Statement 1: If it is a triangle, then it has sharp corners. TRUE Statement 2: If is has sharp corners, then it is a triangle. FALSE Counterexample: Depending on what is meant by sharp, a square could be considered to have sharp corners. The statement is not a good definition because it uses the imprecise word sharp, and it is not reversible.

Biconditionals and Definitions Is the following statement a good definition? Explain. A square is a figure with four right angles.

The Truth Table for a Biconditional p q T F

The Truth Table for a Biconditional p q T F

The Truth Table for a Biconditional p q T F

The Truth Table for a Biconditional p q T F

Who wants to answer the Guiding question? What is a biconditional?