Conditional Statements, Biconditionals, and Deductive Reasoning Part 2 Conditional Statements Conditional Statements • A conditional is an If – then statement – p q (read as if p then q or p implies q) • The Hypothesis is the part p following if • The Conclusion is the part q following then. Identifying the Hypothesis and Conclusion • What is the hypothesis and conclusion of the conditional? – If an animal is a robin, then the animal is a bird • H – An animal is a robin • C – The animal is a bird – If an angle measures 130, then the angle is obtuse • H – An angle measures 130 • C – The angle is obtuse Writing a Conditional • Write the following statement as a conditional – Vertical angles share a vertex • Step 1: Identify the Hypothesis and Conclusion – H: Vertical Angles – C: Share a vertex • Step 2: Write the Conditional – If two angles are vertical, then they share a common vertex – You Try: How can you write “Dolphins are mammals” as a conditional? • If an animal is a dolphin, then it is a mammal Truth Value • The truth value of a conditional is either true or false. • To show a conditional is true, show that every time the hypothesis is true, the conclusion is also true • To show a conditional is false find one counter example for which the hypothesis is true and the conclusion is false Finding the Truth Value of a Conditional • Is this conditional true or false, if it is false find a counter example. – If a women is Hungarian, then she is European. • This is True! – If a number is divisible by 3, then it is odd. • This is false, the number 12 is divisible by three and not odd. – If a month has 28 days than it is February • This false, January has 28 days – If two angles form a linear pair, then they are supplementary • True! Negation • The negation of a statement p is the opposite of that statement, the symbol is ~p and is read “not p” – Example: • The negations of the statement “the sky is blue” is “the sky is not blue” • You use the negation to write statements related to a condition Related Conditional Statements Statement How To Write Example Symbol How to read Conditional Use the given hypothesis and conclusion If m<A = 15, then <A is acute p q If p, then q Converse Exchange the hypothesis and conclusion If <A is acute, then m<A = 15 q p If q, then p Inverse Negate both the hypothesis and conclusion from the conditional If m<A ≠ 15, then <A is not acute ~p ~ q If not p, then not q Contrapositiv Negate both the e hypothesis and conclusion from the converse If <A is not acute, then m<A ≠ 15 ~q ~p If not q, then not p Truth Value Statement Example Truth Value Conditional If m<A = 15, then <A is acute True Converse If <A is acute, then m<A = 15 False Inverse If m<A ≠ 15, then <A is not acute False Contrapositive If <A is not acute, then m<A ≠ 15 True Equivalent Statements have the same truth value, the conditional and contrapositive are equivalent, and are the converse and inverse statements. You Try • Write the Converse, Inverse and Contrapositive statements – IF a vegetable is a carrot, then it contains beta carotene – Converse • If a vegetable contains beta carotene then it is a carrot – False (Spinach has Beta Carotene) – Inverse • If a vegetable is not a carrot then it does not contain beta carotene – False – Contrapositive • If a vegetable does not contain beta carotene then it is not a carrot – True! Part 3 Biconditionals Biconditional • A single true statement that combines a true conditional and its true converse, you can write a biconditional by joining the two parts of each conditional with the phrase if and only if – Symbol: Writing a Biconditional • To write a biconditional first determine if the what is the converse of the following true conditional. If the converse is true then write a biconditional statement – Conditional: If the sum of the measure of two angles is 180, then the two angles are supplementary – Converse: If two angles are supplementary, then the sum of the measures of the two angles is 180 • Biconditional: – Two angles are supplementary if and only if the sum of the measures of the two angles is 180 You Try • What is the converse of the following conditional, if the converse is true write a biconditional statement • If two angles have equal measures, then the angles are congruent – Converse: If angles are congruent, then they have equal measures • Biconditional – Two angles have equal measures if and only if they are congruent Identifying the conditionals in a Biconditional • What are the two statements that form a biconditional – A ray is an angle bisector if and only if it divides and angle into two congruent angles • Find p and q • P – A ray is an angle bisector • Q – A ray divides an angle into two congruent angles – Conditional: If a ray is an angle bisector, then it divides the angle into two congruent angles – Converse: If a ray divides and angle into two congruent angles, then it is an angle bisector You Try! • What are the two conditionals that form this biconditional? – Two numbers are reciprocals if and only if their product is one. • Conditional: If two numbers are reciprocals, then their product is one • Converse: If two numbers product is one, then they are reciprocals. Part 4 Deductive Reasoning Deductive Reasoning • Also called logical reasoning, is the process of reasoning logically from given statements or facts to a conclusion Law of Detachment • If the hypothesis of a true conditional is true, then the conclusion is true • If p then q is true and p is true, then q is true Using the law of detachment • What can you conclude from the given true statements? – If a student gets an A on a final exam, then the student will pass the course. Felicia got an A on her history Final • Felicia will pass the course – If a ray divided an angle into two congruent angles, then the ray is an angle bisector. Ray RS divides <ARB so that <ARS ≅ <SRB • Ray RS is an angle bisector More Examples – If two angles are adjacent, then they share a common vertex. <1 and <2 share a common vertex. • Since the second statement does not match the hypothesis then we can not conclude anything – If there is lightning, then it is not safe to be out in the open. Marla sees lightning from the soccer field. • It is not safe to be out in the open – If a figure is a square, then its sides have equal lengths, figure ABCD has sides of equal length. • We can not conclude this is a square because out statement matched the conclusion not the hypothesis Law of Syllogism • Allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of another statement – If p q is true – And q r is true – Then p r is true Using the law of Syllogism • What can you conclude from the given information? • If a figure is a square, then the figure is a rectangle. If a figure is a rectangle than the figure has four sides – If a figure is a square then it has four sides • If you do gymnastics, then you are flexible. If you do ballet then you are flexible. – Each conclusion is the same so we can not use the law of syllogism and can conclude nothing More Examples • If a whole number ends in 0, then it is divisible by 10. If a whole number is divisible by 10, than it is divisible by 5. – If a whole number ends in zero is it divisible by 5 • If Ray AB and Ray AD are opposite rays, then the two rays form a straight angle. If two rays are opposite rays, then the two rays form a straight angle. – The hypothesis and conclusion matches so we can make no further conclusions The END!