Chapter 2 2 3 Deductive Reasoning Determine what

Chapter 2 2. 3 Deductive Reasoning

Determine what follows from these two statements • If I have reached the party to whom I am speaking, then I have dialed correctly I have indeed reached the party to whom I am speaking I have dialed correctly • If the Jolly Green Giant started turning blue, he should put on a sweater The Jolly Green Giant has not started turning blue NOTHING FOLLOWS

Determine what follows from these two statements • If you see spots in front of your eyes, then you are looking at a leopard You’re looking at a leopard NOTHING FOLLOWS • If you forgot your pencil, then you may borrow one of mine. You forgot your pencil You may borrow one of mine

Symbolic Notation • Conditional: If a then b • Conditional Symbolized: a b – In English we would say “a implies b” • Example: If the sun is out then the weather is good – the sun is out the weather is good • s g

Symbolic Notation • Biconditional: a if and only if b • Biconditional Symbolized: a b – In English we would say “a if and only if b” • Example: The sun is out if and only if the weather is good – The sun is out The weather is good • s g

Symbolic Notation • Negation: not • Negation Symbolized: ~ • Example: The sun is out – Negation: The sun is not out – Symbolized: ~s

Write the symbolic statement in words p: The value of x is 4 q: 3 x + 2 = 14 1. ~p • The value of x is not 4 2. ~q • 3 x + 2 14 3. q p • If 3 x + 2 = 14 then the value of x is 4

Write the symbolic statement in words p: The value of x is 4 q: 3 x + 2 = 14 4. ~q ~p • If 3 x + 2 14 then the value of x is not 4 5. p q • If the value of x is 4 then 3 x + 2 = 14 6. ~p ~q • If the value of x is not 4 the 3 x + 2 14

Determine what follows from these two statements 7. If an angles measures more than 90 then it is not acute. M ABC = 120 ABC is not acute 9. If you order apple pie, then it will be served with ice cream. Matthew ordered apple pie Mathew was served ice cream

Determine what follows from these two statements 10. If you wear school colors, then you have school spirit. If you have school spirit, then the team feels great. If you wear school colors, the team will feel great. 11. If you eat to much turkey then you will get sick Kinsley got sick Kinsley ate to much turkey.

The law of detachment • If p q is a true conditional statement and p is true , then q is true – If the class is boring, then the students will stop paying attention The class is boring The student are not paying attention – –

The law of syllogism • If p q and q r then p r is a true conditional statement 12. If 2 is acute, then 3 is obtuse If 3 is obtuse, then 4 is acute If 2 is acute, then 4 is acute

Determine what follows from these two statements 13. All 45 angles are A B A and B are 45° Angles Counter example: The angle could be 60°

More Symbolism • Equivalent statements Conditional Statement p r Contrapositive ~r ~p • Example • If the car will start, then the battery is charged • If the battery is not charged, then the car will not start

More Symbolism • Equivalent statements Converse r p Inverse ~p ~r • Example • If the battery is charged, then the car will start • If the car will not start, then the battery is not charged

1. If Susan screams then the dog will run away If the dog licks Susan, then Susan will scream If Carl tells the dog to “kiss”, then the dog will lick Susan Carl tells the dog to “kiss”. Contapositive of #3 14. If the dog did not lick Susan, then Carl did not tell the dog to “kiss”. Inverse of #2 15. If the dog does not lick Susan, then Susan will not scream. Converse of #1 16. If the Dog runs away, then Susan screamed.

1. If Susan screams then the dog will run away If the dog licks Susan, then Susan will scream If Carl tells the dog to “kiss”, then the dog will lick Susan Carl tells the dog to “kiss”. 16. Appropriate Order? 3, 2, 1, 4 17. Did the dog lick Susan? Yes 18. Did the dog run away? Yes