Inductive and Deductive Reasoning
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Transcript of Inductive and Deductive Reasoning
MATH FOUNDATIONS 11
Inductive and Deductive Reasoning
Let’s play a little game
Pick the number of days per week that you like to eat chocolate
Multiply this number by 2Now, add 5Multiply this new number by 50
If you’ve already had your birthday this year, add 1764, if not, add 1763
Now, subtract the four digit year that you were bornWhat do you notice about your answer?The first digit is your original #, and the last two digits
your age
Here’s a Fun Quiz
Think of a number from 1 -10Multiply that number by 9If the number is 2 digits, add the digits togetherNow, subtract 5Convert your number to a letter: A=1, B=2, C=3,
D=4, …Think of a country that starts with that letterRemember the last letter of the name of your country,
and think of an animal that starts with that letterRemember the last letter of your animal, and think of
a fruit that starts with that letterAre you thinking about …..
Orange Kangaroo from Denmark?
Proof that 2 = 1
Let a=bSo, a2 = ab
a2 - b2 = ab - b2 (a-b)(a+b) = b(a-b)
(a+b) = bb + b = b
2b = b2 = 1
4 – 9’s
Apparently, all the numbers from 0 to 100 can be created from four nines
Ex: 9 + 9 – 9 - 9 = 0Ex: 9/9 + 9 – 9 = 1Ex 9/9 + 9/9 = 2
Are there any numbers that can’t be done?
1.1 Inductive ReasoningInductive reasoning – drawing a general
conclusion by observing patterns and identifying properties in specific examples.
Conjecture – A testable expression that is based on available evidence but is not yet proved.
Example
1(8) + 1 = 912(8) + 2 = 98
123(8) + 3 = 9871234(8) +4 = 9876
12345(8) + 5 = ____
Does this pattern continue forever?
No!
Verbal example
This swan is whiteI’ve seen 100 white swans
All swans are white
Is this always true?
No!
Assignment – Page 12 # 3 – 17 odd
Section 1.2
Section 1.2
Section 1.2
Section 1.2
1.3 Using Reasoning to Find a counterexample to a Conjecture
A conjecture can be proven false by finding a counter example
Statement: A number that is not negative is positive
Statement: A number that is not negative is positive
Counter example :
This statement is proven false by finding a counter example
o
Statement: All prime numbers are odd
Statement: All prime numbers are odd
Counter example :
This statement is proven false by finding a counter example
2
Statement: All NBA basketball players are tall
Statement: All NBA basketball players are tall
Counter example :
This statement is proven false by finding a counter example
Nate Robinson(5’7”)
Statement: The square root of a positive number is always less than the number itself
Statement: The square root of a number is always less than the number itself
Counter example :
This statement is proven false by finding a counter example
1
Statement: As you travel north, the climate gets colder
Statement: As you travel north, the climate gets colder
Counter example :
This statement is proven false by finding a counter example
Southern hemisphere
Statement: The sum of 2 numbers is always greater
than the greater of the 2 numbers.
Statement: The sum of 2 numbers is always greater than either of the 2 numbers.
Counterexample: -3 + -5
This statement is proven false by finding a counter example
Assignment
Page 22 #2 – 9, 12, 17
1.4 Deductive Reasoning
Deductive reasoning is a process where we draw conclusions using logic that is based on facts we accept as true
A conjecture is proved true only when it is true for every case. This is done by creating a proof for general cases.
Example
All poodles are dogsAll dogs are mammals
Example
All poodles are dogsAll dogs are mammals
All poodles are mammals
Example
Lynn is a math studentAll math students have a calculator
Example
Lynn is a math studentAll math students have a calculator
Lynn has a calculator
Integer Property Proof
Every integer is either ODD or EVEN
Let j, k be random integers
Need an even integer?Let = an even integer
Need an odd integer?Let = an odd integer
“Let” statements
Need consecutive integers? Let k = 1st integer = 2nd integer = 3rd integer
Need consecutive even integers? Let = 1st even integer Let = 2nd even integer
Need consecutive odd integers? Let =1st odd integer Let = 2nd odd integer
Example
Prove that the square of any odd integer is odd
Let = 1st odd integer
Example
Prove that the product of any 2 even integers is even
Let 2k = 1st even integerLet 2j = 2nd even integer
Product = (1st integer)(2nd integer) = (2k)(2j) = 4kj
This number is divisible by 2, therefore it is EVEN
AssignmentPage 31 # 1, 2, 4 – 8, 10, 12,
17, 20
1.5 Proofs that are not valid
Jean Chretien: http://www.youtube.com/watch?v=aX6XMIldkRU
1.5 Invalid Proofs
Invalid Proof – a proof that contains an error in reasoning or that contains invalid assumptions.
Circular Reasoning – an argument that is incorrect because it makes use of the conclusion to be proved.
1.6 Reasoning to Solve Problems
If 3 cats can catch 3 mice in 3 minutes, how long will it take 100 cats to catch 100 mice?
1.6 Reasoning to Solve Problems
A bottle and a cork together cost $1.06. The bottle cost $1 more than the cork. How much does the cork cost?
1.6 Reasoning to Solve Problems
Assuming both players to be intelligent, who started this game of X's and O's, player X or player O? Explain.
1.6 Reasoning to Solve Problems
Suppose you are lost in the woods for hours. You suddenly come upon a cabin. In the cabin you find a lantern, a candle, a wood stove with wood in it , and a match. What do you light first?
1.6 Reasoning to Solve Problems
A man went to town one day with $5 in his pocket, but returned in the evening with $15. He bought a hat at the men's furnishings store and some meat at the meat market. Then he had his eyes tested for glasses. Now, this man got paid every Thursday by check, and the banks in the town are open on Tuesday, Friday, and Saturday only. The eye doctor does not keep his office open on Saturday, and the meat market is not open on Thursday or Friday. What day did the man go to town?
Tuesday
1.6 Reasoning to Solve ProblemsMr. Jones one day got off the train in Chicago
and while passing through the station met a friend he had not seen in years. With his friend was a little girl. "Well, I certainly am glad to see you," said Mr. Jones. "Same here," said his friend. "Since I last saw you I've been married--to someone you never knew. This is my little girl." "I'm glad to meet you," said Mr. Jones. "What's your name? "It's the same as my mother's," answered the little girl. "Oh, then your name is Anne," said Mr. Jones. How did he know?
His friend was a lady, named Anne
There are three light bulbs in a room, and three light switches outside the room. You are outside, and want to match up which switch goes with which light bulb. You can only travel into the room once, and cannot come back in again. You can do anything you want upon entering the room. How can you set the situation so that you will know which switch goes with which light bulb?
Turn two of the light switches on. Wait a while, and then turn one of them off, then quickly enter the room. One light will be on, one off, and one off but still hot.
1.6 Reasoning to Solve Problems
Pepsi Problem – quite hard
A man has a 12 L jug of pepsi, and wants to split it in half, but only has a 8 L and a 5 L jug. How can he do it?
Fill the 8 L jug (then we have (4,8,0) in the 12 L, 8 L, and 5 L)Then fill the 5 L with the 8 L (4,3,5). Pour the five back into the 12 (9,3,0). Then transfer the 3 in the 8 L jug into the 5 L jug (9,0,3). Fill the 8 L with the 12 L (1,8,3). Then Finish filling the 5 L with the 8 L (1,6,5). Finally, pour the 5 L back into the 12 L bottle, for (6,6,0).
1.6 Reasoning to Solve Problems
Page 49 #
1.7 Analyzing Puzzles and Games
The following is a game for two players. Place a pile of 20 pennies on your desk. Determine the starting player. Players alternate turns removing 1 or 2
pennies per turn from the pile. The player to remove the last penny is the
winner.
Play The Game20 pennies.notebook
1.6 Reasoning to Solve Problems
Assuming both players to be intelligent, who started this game of X's and O's, player X or player O? Explain.
O started
1.7 Analyzing Puzzles and Games
Handout: Logic Puzzles, Sudokuhttp://www.sudoku9x9.com/