1. Deductive Reasoning Geometry Chapter 2, Section 3

2. Symbolic Notation for statements Statements can be represented by symbols Example: Statement: If the sun is out, then the weather is good p: the sun is out q: the weather is good If p, then q or p  q Example Converse: If the weather is good, then the sun is out If q, then p or q  p On Your Own:  Define the hypothesis and conclusion of the following statement with letters.  Write the statement and its converse in symbolic form.  If the sky is clear tomorrow morning, then I’ll go for a run.  r: ___________________  s: ___________________  Statement : ____  ____,  Converse: ____  ____

3. Biconditional Statement: use this symbol ↔ Example Biconditional Statement: The weather is good if and only if the sun is out p: the sun is out q: the weather is good P if and only if q, or q ↔ p Symbolic Notation for statements

4. Negation: uses this symbol: ~ ~p is read not p Statement: p  q Inverse: ~p  ~q Contrapositive: ~q  ~p On Your Own: For the statement below, first define the hypothesis and conclusion in symbols then write the converse, inverse and contrapositive in symbols. Statement: If the sky is clear tomorrow morning, then I’ll go for a run. r: ___________________________ s: ___________________________ Statement : ___  ___, Converse: ___  ___ Inverse: ~ ___  ~ ___ Contrapositive: ~ ___  ~ ____ Symbolic Notation for statements

5. In Preparation Open your browsers to maps.google.com Type in Dublin select Dublin, Ireland from the drop down Menu Click the satellite button on the top right of the map portion of the page. Repeat the same procedure for Beijing, China and Cairo, Egypt Close your laptop

6. Notes Deductive Reasoning: uses facts, definitions, and true statements whether assumed or proved to come to conclusions. Law of Detachment: says that if an if-then statement is true and its hypothesis is true, then its conclusion must also be true.  If p  q is true and p is true then q is true  Example:  True Statement: If you over mix your biscuit dough, then it will not rise.  From the law of detachment, I can be assured that my biscuits will be flat and hard if I over mix the dough.

7. On your own: Use the law of detachment to come up with a conclusion If I visit Germany, then I’ll have to learn to eat sour kraut. I’m visiting Prague this summer. Is the hypothesis satisfied? Is it true? What can you conclude? ________________________ What if I visit Frankfurt?____________________ If I have to learn to eat sour kraut, does that mean I’m going to Germany?_________________________ Confirmation of the conclusion doesn’t ensure that the hypothesis is true. The point: the hypothesis must be true for the conclusion to be true

8. Notes Law of Syllogism: says  If p  q is true and q  r is true, then p  r is true also  It’s like a road that gets you to your destination  Example:  True Statement 1: If I get into the pool, then I have to shower first.  True Statement 2: If I have to shower first, then I will be cold before I’m even in the water.  It is horrible rushing to the pool after taking that cold shower isn’t it!

9. On your own: Use the law of syllogism to answer this question  If I want to fly to Hamburg, then I have to stop in either London or Munich  If I stop in Munich, then I must see Neuschwanstein. I have always wanted to see the most famous of Europe’s castles.  On my way to Hamburg this spring, will I get my wish to see Neuschwanstein?__________________ Was there a link between one if-then statement and the next?__________________ _______________________ How could I have rephrased the second statement to make it so a conclusion could be reached?________________ The Point: There has to be a link between the two statements, and you have to proceed from hypothesis to conclusion in your reasoning.

10. Lewis Carroll: Deductive Reasoning Activity Write the statements symbolically as if-then statements, along with their contrapositives, and then string together the statements that match up to arrive at a final conclusion. 1. My saucepans are the only things I have that are made of tin. 2. I find all your presents very useful. 3. None of my saucepans are of the slightest use. p: They are my saucepans q: they are made of tin and mine r: They are presents from you s: I find them very useful r  s; s  ~p; ~p  ~q so r  ~q If They are presents from you, then they are not made of tin q  p; p  ~s; ~s  ~r so q  ~r If they are made of tin, then they are not presents from you! How are these two statements related? OriginalContrapositive 1 st Sentence: q  p~p  ~q 2 nd Sentence: r  s~s  ~r 3 rd Sentence: p  ~ss  ~p

11. Try one on your own: Write the statements symbolically as if-then statements, along with their contrapositives, and then string together the statements that match up to arrive at a final conclusion. No potatoes of mine, that are new, have been boiled. All my potatoes in this dish are fit to eat. No unboiled potatoes of mine are fit to eat. No ducks waltz. No officers ever decline to waltz. All my poultry are ducks. Every one who is sane can do Logic. No lunatics are fit to serve on a jury. None of your sons can do logic.