UNIT 2 Methods of Proof and Logic Kitaboo

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Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . Reasoning Arguments and Proof, When you heat tightly sealed food in the microwave oven pressure builds. Objectives, Lesson Objectives and the lid pops off the container or the food explodes out of the wrapper You. , Define and identify tools quickly discover that if you continue to seal the food tightly when you heat it . for informal reasoning you will have a yucky mess . such as observation Reasoning in which you make general assumptions from specific. measurement and observations is called induction When you use deductive reasoning you. experimentation progress in the opposite direction from the general to the specific You. collect observations and test your hypotheses with thorough information and. , Describe the properties, statistics to confirm or disprove your original assumptions . of proofs , Induction like the way you learn not to heat tightly sealed food in the.
, Identify premises microwave is less exact and more exploratory Deduction is controlled. and conclusions in an and tests hypotheses through previously established properties Deduction is. argument proof induction is not , , Identify and develop. syllogisms and other Keywords, valid arguments argument conclusion. deductive reasoning diagonals, premise proof, syllogism valid argument. Reasoning, Observations can be used to draw conclusions You can use a table to keep.
track of observations and see if a pattern emerges Look at the polygons. below As you look from left to right the number of sides in each polygon. increases by one , The red lines are diagonals Diagonals are segments that connect two. vertices of a polygon and do not lie along any side of the polygon For each. polygon above every diagonal that can be drawn from one vertex is shown The. triangle has no diagonals because any line connecting two vertices would lie. along a side of the triangle The numbers of sides and diagonals are listed in the. table below , Number of Sides of Polygon n 3 4 5 6 7 8. Number of Diagonals 0 1 2 3 4 5, Reasoning arguments and Proof 33. Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . If you study the table for a bit you will see that there is a relationship. between the number of sides of a polygon and the number of diagonals that. can be drawn from one vertex of a polygon So from a simple observation . you can conclude that the number of diagonals that can be drawn from a. single vertex is three less than the number of sides or n 3 . The conclusion about the number of diagonals was based on observation. and studying of patterns You can also derive conclusions on the basis of. measurement and experimentation For example if you measure both. diagonals of any rectangle you will find that their lengths are always equal . It is important to note that while observation measurement and. experimentation can lead you to useful assumptions those tools are not. considered proof Mathematicians rely on deductive reasoning which uses. previously proven or accepted properties to reach formal conclusions . Logic and Proof, An argument is a set of statements called premises which are used to reach. a conclusion Both the premises and the conclusion are considered to be part. of the argument , A syllogism is a special kind of logical argument It always contains two.
premises and a conclusion Syllogisms have the following form . Premise If a then b , Premise If b then c , Conclusion Therefore if a then c . Example , If Fido is hungry in the morning then he barks . If Fido barks then Jenny wakes up , Therefore if Fido is hungry in the morning then Jenny wakes up . The validity of an argument is based on the structure of the argument . RECONNECT TO The syllogism above is a type of valid argument In a valid argument if the. THE BIG IDEA premises are all true then the conclusion must also be true . Remember Deductive The following is an example of an invalid argument The structure of the. reasoning enables us to argument is faulty Notice that both premises can be true and the conclusion. derive true conclusions from can be false , statements accepted as true . If you are on a baseball team then you wear a red hat . If you are a fireman then you wear a red hat , Therefore if you are on a baseball team then you are a fireman .
34 Unit 2 Methods of proof and logic, Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . Proofs, Earlier in the book proof was defined as a clear logical structure of reasoning. that begins from accepted ideas and proceeds through logic to reach a. conclusion In other words a proof uses deductive reasoning In a proof only. valid arguments are used so the conclusions must be valid Forms of valid. arguments and different types of formal proofs will be shown throughout this. unit , Summary, O, bservation measurement and experiments are useful but they are not. methods of proof A formal proof uses deductive reasoning to reach a. conclusion , A, n argument is a set of statements that are made up of a set of premises. and a conclusion The premises provide support for the conclusion . A, syllogism is a special kind of logical argument that contains two premises.
and one conclusion , I n a valid argument the premises cannot be true and the conclusion false. at the same time , Reasoning arguments and Proof 35. Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . Conditional Statements, A person who lives in Goodmath must be at least 17 years old to obtain. Objectives, Lesson Objectives a full driver s license Nathan who lives in Goodmath is over the age of. , Identify and formtypes 17 Although the two previous statements are true can you conclude that.
D efine the three, conditional Nathan has a driver s license No there is not enough evidence Nathan. of symmetrystatements , reflection , parts of conditional probably has a driver s license but you cannot be certain of it Nathan s. rotation and translation , statements and forms of sister Julie has a driver s license and lives in Goodmath Can you presume. , Identify lines of symmetry that she is over the age of 17 Yes this deduction based on previously known. conditional statements , , Identify reflection rotation truths does follow from the given information .
, Create and interpret truth, and translation symmetry Be aware that although deductive reasoning seems to be an easy concept. tables for conditional it s possible for someone to be tricked into faulty reasoning There is a. in different figures , statements definite difference between an untrue hypothesis and an argument with. , Create or interpret Euler flawed logic , diagrams that model. conditional statements Keywords, , Use the Law of conclusion conditional statement. Contrapositives and the contrapositive converse, If Then Transitive Property hypothesis inverse.
logical chain statement, truth functionally equivalent. Conditional Statements, A statement is a sentence that is either true or false A conditional statement. is a statement that has two parts The first part begins with the word if and the. second part begins with the word then The hypothesis includes the words. following if and the conclusion includes the words following then In the. conditional statement If it is sunny then I will mow the grass the words in. blue are the hypothesis and the words in red are the conclusion . Forms of Conditional Statements, MemoryTip By rearranging and negating the hypothesis and the conclusion you can. form the converse inverse and contrapositive of a conditional statement . The prefix in often means not , The converse switches the hypothesis and the conclusion The inverse. For instance insane means not, negates or takes the opposite of both the hypothesis and the conclusion .
sane , The contrapositive both switches and negates the hypothesis and the. conclusion , conditional statements 37, Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . Conditional Statement If it is sunny then I will mow the grass . Converse If I mow the grass then it is sunny , Inverse If it is not sunny then I will not mow the. grass , Contrapositive If I do not mow the grass then it is not. sunny , You can use letters and symbols to represent the forms of these statements .
In logic the letters p and q are usually used When you use a variable in. mathematics you are using the variable to represent a number In the same. way the letters p and q represent a group of words . You can use the symbol to mean implies in a conditional statement . When you write p q you are saying that p implies q which is the same as. if p then q The symbol is read as not , Conditional Statement If p then q p q. Converse If q then p q p, Inverse If p then q p q. Contrapositive If q then p q p, Both the hypothesis and the conclusion can be either true or false This is. called the truth value of each part The truth values of those clauses determine. whether the entire conditional is either true or false . Truth Tables, Truth tables are an organized way to look at the possible truth values of an. expression The truth table for a conditional statement lists every possibility. of truth values for the hypothesis conclusion and statement where T means. true and F means false , p q p q Conditional Statement The truth table for a conditional statement is.
shown to the left Notice that the only time p q is false is if the hypothesis. T T T, is true and the conclusion is false In the mowing example this happens when. T F F it is sunny and I do not cut the grass When both the hypothesis and the. F T T conclusion are true the conditional statement is true In other words if it is. F F T sunny then I cut the grass If the hypothesis is false the conditional statement. is true regardless of the conclusion You don t know what I will do if it isn t. sunny so the conditional statement is true by default . p q q p Converse When looking at the truth table for the converse remember that. q is now the hypothesis and p is the conclusion So the converse is only false. T T T, when q is true and p is false , T F T, F T F. F F T, 38 Unit 2 Methods of proof and logic, Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . Inverse To make the truth table for the inverse negate each clause first . Notice that the truth values of p and q are opposite those for p and q The. inverse is only false when p is false and q is true . p q p q p q, T T F F T, T F F T T, F T T F F, F F T T T. Contrapositive To make the truth table for the contrapositive negate p and q. and then remember to switch the order so q is the hypothesis The conclusion. is only false when p is true and q is false , p q p q q p.
T T F F T, T F F T F, F T T F T, F F T T T, Look at the the far right column in each table The converse and the. inverse columns match Likewise the original conditional statement and. the contrapositive columns match Those pairs of statements are truth . functionally equivalent they are either both true or both false . Most important is that the conditional statement and the contrapositive. are equivalent Because you can write any statement in if then form this. will actually double the number of postulates and theorems you have It also. means that if you prove one of the statements then you have also proven the. other , Law of Contrapositives If you prove the contrapositive of a conditional. statement then you also prove the conditional statement and vice versa . Euler Diagrams, Euler diagrams are used to show how the parts of conditional statements. are related The inner circle represents the hypothesis and the outer circle. represents the conclusion If a point lies in the inner circle then it also lies in. the outer circle Or said another way If p then q . q, p, p q, conditional statements 39, Copyright 2006 K12 Inc All rights reserved This material may not be reproduced in whole or in part including illustrations without the express prior written consent of K12 Inc . The Euler diagram below represents the conditional statement which is If it. is sunny then I will mow the grass If a point lies in the it is sunny circle . then it also lies in the I will mow the grass circl. 2unit MEtHODS OF PROOF anD lOgic 31 lic and reasoning will help you in many parts of your life og in talk ing with friends listening to political speeches and learning math science and even English literature and composition you need to be able to understand and create convincing arguments

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