Topic 2Deductive Reasoning

Unit 1 Topic 2

ExploreMany books and movies are based on the fictional detective Sherlock Holmes. Holmes uses logical reasoning to deduce the solution to difficult cases. A favourite quote of Holmes is "When you have eliminated the impossible, whatever remains, however improbable, must be the truth".

Try solving the following case:

• Sherlock Holmes came to a house where a woman lived with her three sons. "What are the ages of your 3 sons?" asked Holmes. "The product of their ages is 72 and the sum of their ages is the same as my house number" replied the woman.

• Sherlock Holmes ran to the door and looked at the house number. "I still can't tell," he complained. The woman replied, "Oh that's right, I forgot to tell you that the oldest son loves chocolate milk."

• Sherlock Holmes promptly wrote down the ages of the 3 sons. How old are they?

A Hint:

•Make a list of all possible ages for the sons.

Solution

When you make a list of all possible combinations for the ages of the 3 sons, you get the following:

1st son 2nd son 3rd son Sum

1 1 72 74

1 2 36 39

1 3 24 28

1 4 18 23

1 6 12 19

1 8 9 18

2 2 18 22

2 3 12 17

2 4 9 15

2 6 6 14

3 3 8 14

3 4 6 13

SolutionSince Holmes didn’t have enough information after checking the house number, we have to assume that the sum of the children’s ages (and the house number) is 14.

1st child 2nd child 3rd child Sum

1 1 72 74

1 2 36 39

1 3 24 28

1 4 18 23

1 6 12 19

1 8 9 18

2 2 18 22

2 3 12 17

2 4 9 15

2 6 6 14

3 3 8 14

3 4 6 13

Solution

Once the woman tells Holmes that her oldest son likes chocolate milk, we know that there must be an ‘oldest.’

2 6 6 14

3 3 8 14

The sons must be 3, 3, and 8.

Information• Remember that inductive reasoning is a type

of reasoning in which a statement or conclusion is developed based on patterns or observations.

• Deductive reasoning is where a statement or conclusion is developed based on true statements.

InformationConsider the following conjecture: The sum of four consecutive numbers is equal to four times the first number plus 6.

• Show this to be true with both inductive reasoning and deductive reasoning

1 2 3 4 10

4(1) 6 10

5 6 7 8 26

4(5) 6 26

Inductively:

or

InformationConsider the following conjecture: The sum of four consecutive numbers is equal to four times the first number plus 6.

• Show this to be true with both inductive reasoning and deductive reasoning

( 1) ( 2) ( 3)

1 2 3

4 6

n n n n

n n n n

n

Deductively:

Why is deductive reasoning more reliable than inductive reasoning?

It is proven to be true for all values, with statements we already know are true.

Information

In order to come to conclusions deductively, we often use algebra. In order to do this, we must define some general terms.  How do we write the following in general terms? 

a) an integerb) two consecutive numbersc) an even numberd) two consecutive squarese) an odd number

 

Information

a) an integer n

b) consecutive numbers Since an integer is n, consecutive integers are

n, n+1, n+2, n+3, etc...

c) an even number Since an even number is a number that is

divisible by 2, we can define an even number as 2n.

 

Information

d) two consecutive squares A squared number is n2, so consecutive

squares are n2, (n+1)2, (n+2)2, etc...

e) an odd number Since an odd number always follows an

even number, add one to the definition of an even number: 2n+1

 

Information

Example 1Proving by Deductive Reasoning

Use deductive reasoning to prove the following statements.

a) When five consecutive integers are added together, the sum is always 5 times the median of the numbers. 

b) The difference between consecutive perfect squares is an odd number.

c) When two odd numbers are added, the sum is always even.

Try this on your own first!!!!

Example 1a: Solution

Example 1a: Video Solution – click here

Example 1b: Solution

Example 1b: Video Solution – click here

Example 1: Solution

c) When two odd numbers are added, the sum is always even.

Define 2 distinct odd numbers: 2 1n and

2 1m Add them together: 2 1 2 1n m Add them together and simplify:

2 1 2 12 2 22( 1)n mn mn mSince 2 can be

factored out, the expression is, by definition, even.

Example 2Writing Conclusions

Deductive reasoning can also be used to make a valid conclusion based on given statements.

Write a valid conclusion which can be deduced from the statements.

a) Every whole number is an integer. Six is a whole number.b) Water freezes below 0˚C. The temperature is -15˚C.c) All quadrilaterals have four vertices. A parallelogram is a

quadrilateral.d) All dogs are mammals. All mammals are vertebrates.

Shaggy is a dog.e) All runners train on a daily basis. Jen is a runner.

Try this on your own first!!!!

Example 2: Solution

a) Every whole number is an integer. Six is a whole number.

b) Water freezes below 0˚C. The temperature is -15˚C.

c) All quadrilaterals have four vertices. A parallelogram is a quadrilateral.

d) All dogs are mammals. All mammals are vertebrates. Shaggy is a dog.

e) All runners train on a daily basis. Jen is a runner.

Six is an integer

The water is frozen.

A parallelogram has four vertices

Shaggy is a mammal.

Jen trains on a daily basis.

Example 3General Cases

Try this on your own first!!!!

a)Complete the following table. Start by choosing any two numbers, labelled in the table as Value 1 and Value 2.

Value 1 Value 2

Choose a number.

Double it.

Add 5.

Add the original number.

Add 7.

Divide by 3.

Subtract the original number.

b) State a conjecture about the final result.  c) Using deductive reasoning, prove your conjecture for the general case.

Example 3: Solution

a)Choose any values, first. I choose 6 and 7. Then, complete the table, step by step for each number.

Value 1 Value 2

Choose a number. 6 7

Double it. 12 14

Add 5. 17 19

Add the original number. 23 26

Add 7. 30 33

Divide by 3. 10 11

Subtract the original number.

4 4

b) Conjecture: When you follow the steps, the answer will be 4.

Example 3: Solution

c) To solve this deductively, complete the same chart, but use letters to represent the numbers to prove the general case.

General Case

Choose a number. x

Double it. 2x

Add 5. 2x+5

Add the original number. 2x+5+x=3x+5

Add 7. 3x+5+7=3x+12

Divide by 3. (3x+12)/3=x+4

Subtract the original number. x+4-x = 4

Need to Know:

• Deductive reasoning starts with general theories that are true and, through logical reasoning, arrives at a specific conclusion.

• A conjecture has been proven only when it has been proved true for every possible case or example. This is done through general cases.

You’re ready! Try the homework from this section.