2.3+Deductive+Reasoning

=Chapter 2 Section 3= By Sung Jin Kim and Stephanie Stein
 * Summary**- Here, you will earn of proofs and deductive reasoning. You will learn about the parts of proofs, types of proofs, and notation of proofs. You will also learn about deductive reasoning which is figuring one thing out from learning another thing. That can be used in every day life and geometry.

Symbolic Notation
A proof consists of two parts: the hypothesis and conclusion. A proof is formatted in an "if...then..." format. This will be explained further on.

There are two symbols the show the parts of the proof //p//: the hypothesis //q//: the conclusion The symbols show which part of the proof is which by covering the corresponding parts underneath the proof.

The hypothesis show the first part of the sequence in a proof. It is the "if" part of the proof.

Ex: __If I bake cookies,__ then I will eat them later. // ................. p The conclusion shows what will happen if the hypothesis comes true.

Ex: If I bake cookies, __then, I will eat them later.__ // // ................................................ q// Another way symbolic notation comes into play is like this //p//--->//q.

That is the symbolic notation show a conditional statement. To learn what a conditional statement is and other forms of proofs are, please go to section 2.1 and 2.2.

Conditional: p ---> q If an angle is right, then it is 90°. ............. p............|.......q........... Inverse: ~ p --->~ q If an angle is not right, then it is not 90°. ................ p ...............|.......... q............. Converse: q//--->//p If an angle is 90°, then it is right. ........... p ...........|.......... q.......... Contrapositive: ~q--->~p If an angle is not 90°, then it is not right. ............. p ................|.......... q//...............

Laws of Logic
Inductive and Deductive reasoning are two laws of logic. We will focus on deductive reasoning.

Deductive reasoning is getting a fact from another fact. Ex: You know: I ate the cookies. You figure out: The cookies are gone.

That was the law of detachment. That is when you directly know that because one thing happened, another thing happened. There is another called the law of syllogism.

Ex: You know: I ate the cookies You know: If I ate the cookies, then, The cookies are gone. You know: If the cookies are gone, then, mother will be mad. You figure out: Because the cookies are gone, mother will be made.

In the law of syllogism, you take out the middle part of a sequence of events and just connect the first and the last events together. **Practice** 1. Explain the law of Syllogism and using the law of Syllogism, explain what you would ultimately get if you had a recycled water bottle. You get a nickel for every five pennies and five pennies for every recycled water bottle. 2. Explain the law of Detachment and using the law of Detachment, explain what would happen if lose your bike. A search would occur //if// the bike was lost. 3. Using the correct notations, show which part of this sentence is the hypothesis and the conclusion: If I can run ten miles, then, I can take part in the race.

Answers: 1. The law of syllogism is the law stating that if an event causes another which causes another, then the first event causes the last event; a nickel 2. The law of detachment is the law stating that if an even causes another event, then the second event will happen if the first does; a search would occur. 3. //p//: If I can run ten miles //q//: then, I can take part in the race.

=More Info.= http://zimmer.csufresno.edu/%7Elarryc/proofs/proofs.html http://www.sparknotes.com/math/geometry3/geometricproofs