PROOF,CONTRAPOSITIVE,CONTRADICTION
MATH CIRCLE(BEGINNERS)04/29/2012
P=⇒Q means“If P then Q.”For example when P is the statement“Annika drives a car”,and Q is the statement“Annika has a driver’s licen”,then P=⇒Q is the statement“If Annika drives a car,then she has a driver’s licen.”This kind of statement is called an implication,or a conditional statement.
We will u¬P to reprent the statement“P is fal.”With P and Q as above,¬P reprents the statement“Annika does not have a driver’s licen.”
For each of the following statements,write down the statement which is its logical negation:
(a)“Clint is the worst Math Circle instructor.”
(b)“There exists an even number greater than or equal to4which is not the sum of two prime numbers.”
(c)“All swans are white.”
(d)“London is the capital of England,and Sydney is the capital of Australia.”
(e)“The aren’t the droids you’re looking for.”
Copyright c 2008-2011Olga Radko/Los Angeles Math Circle/UCLA Department of Mathematics.
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(f)“Either I’m going crazy,or there’s an elephantfloating in mid-air in the other corner of the room.”
(g)“Behind every great man,there’s a great woman.”
(h)“If x<10,then x2<100.”
(i)“If it’s not raining,Angela did not bring an umbrella.”
(1)(Pizza)The statement“If Zhubeen eats50pieces of pizza in the morning,he will feel very sick in the afternoon.”is a true statement.
Which of the following is logically equivalent to the above statement?
(a)If Zhubeen does not eat50pieces of pizza in the morning,he will not feel very sick in the afternoon.
(b)If Zhubeen does not feel very sick in the afternoon,he did not eat50pieces of pizza in the morning.
(c)If Zhubeen feels very sick in the afternoon,he ate50pieces of pizza in the morning.
(2)(Warblers)Suppo the following three statements are true:
(S1)Every warbler has either a blue stripe or a red stripe(but not both).
(S2)If a warbler has a red stripe,it sings beautiful songs all day long.
(S3)If a warbler is sad,it does not sing beautiful songs all day long.
Which of the following can we logically conclude?
(a)If a warbler has a blue stripe,it does not sing beautiful songs all day long.
(b)If a warbler is sad,then it has a blue stripe.
(c)If a warbler sings beautiful songs all day long,it has a red stripe.
(d)If a warbler has a blue stripe,then it is sad.
什么是白领
Suppo we know that if statement P is true,then Q and ¬Q are both true–that is,we prove that P =⇒
(Q and ¬Q ).It can’t be the ca that Q and ¬Q are both true,so this means that if P were true,then something fal would also be true.Therefore P cannot be true.
A statement like Q and ¬Q is called a logical contradiction ,and an argument like the above to show P is fal is called a proof by contradiction .
Remember rational versus irrational?A rational number is one that can be written as the ratio of integers (whole numbers).4/7is rational,-34/5is rational,21is rational,and 9.39is Theorem:√2is irrational.Proof:The statement we ultimately want to prove is that √2is irrational,but we will do so by assuming that √2is rational and then using that assumption to prove some contradiction.Assume (for the sake of argument)√2is rational.Then √2=a b for some integers a and b .We can assume a and b are relatively prime.
(3)Why can we assume that?
洗米水If we square both sides of √2=a b
,we get 2=a 2
b
工程保修期
2,which is the same as 2b 2=a 2.
But since the left hand side is even (divisible by 2),the right hand side must be even as well.If a 2is even,then a is even.(Why?)So we can write a =2c for some value c .Then the equation is
2b 2=(2c )2,which leads to
2b 2=4c 2,and then dividing both sides by 2gives网络连接受限
b 2=2
c 2.
But now the right side of the equation is even,which means the left hand side must be even also,and if b 2is even,then b must be even.
(4)I claim this contradicts something we what is the contradiction?
(5)Prove that if x+y is irrational,then at least one of x or y is irrational.(Hint: Contrapositive.)
(6)John is confud about the proof that √
2is irrational.“Why doesn’t the same
line of reasoning also prove that √
4is irrational?”he wonders.Where does the
argument break down?(Write down the same argument as above,except replace √2
with √蓝星花
4and make the obvious changes.When does something go wrong?Show your
assistant.)
(7)Show that there exist two numbers a and b,both irrational(possibly the same number),such that a b(a raid to the b th power)is rational.
(Important Hint:I know √
2is irrational,so if
√
2
孔繁森事迹√
2
is rational,then the statement is
true.What if √
2
√番茄牛肉意面
2
is irrational,though?
(8)Sam is very plead with his most recent purcha,a shiny book called the Com-plete Non-Self-Reference Reference(CNSRR).The ads tell us that this book mentions exactly all the books that doesn’t mention themlves.For instance,Harry Potter and the Sorcerer’s Stone doesn’t mention it
lf,as a book,anywhere in the text,so CNSRR mentions it.On the other hand,America(The Book)does mention itlf,so it does not get mentioned in CNSRR.
When Edanel hears about Sam’s new purcha,he gets worried.“I think there’s a problem with your new book,”he says.What does Edanel mean?What’s the problem?
(9)In a certain small town in Idaho,there is a curious law:“The town barber shaves exactly tho men who do not shave themlves.”Do you e any problem with this law?
>暴力的英文
