Math 20-2
Reasoning: Lesson #3
Invalid Proofs
Objective: By the end of this lesson, you should be able to:
- Determine whether a proof is valid, and justify the reasoning.
- Identify errors in a given proof.
Vocabulary
 Invalid Proof –

Premise –
Some proofs are not valid. This can be for a variety of reasons.
Reason #1: The proof uses inductive rather than deductive reasoning.
Remember: Examples (even LOTS of examples) do not prove a statement.
e.g. 1) Explain the error in this proof:
No one I know has ever gotten 100% in math. Therefore, it’s impossible to get 100% in
math.
Reason #2: The proof contains a faulty premise.
e.g. 2) Explain the error in this proof:
Athletes do not compete in both the Summer and Winter Olympics. Hayley Wickenheiser
has played on the Canadian Women’s hockey team in four Winter Olympics. Therefore,
Hayley Wickenheiser has not competed in the Summer Olympics.
Math 20-2
Reasoning: Lesson #3
Reason #3: The proof contains circular reasoning.

Circular Reasoning –
e.g. 3) Explain why the following are examples of circular reasoning:
a) The reason that stealing is illegal is because it is prohibited by the law.
b) Parent: “It’s time for bed.”
Child: “Why?”
Parent: “Because it’s bedtime.”
c) To prove that 5  5 :
Assume that 5  5
2
2
Square both sides:  5  5
25  25
This is a true statement, so 5  5 must be correct.
Reason #4: Inverse Error
This type of error happens when you assume that if the first premise is false, then the statement
that follows is also false.
e.g. 4) Explain the error in this proof:
If someone is a Canadian citizen, then (s)he is a human being. Queen Elizabeth is not a
Canadian citizen, therefore she is not a human being.
Math 20-2
Reasoning: Lesson #3
Reason #5: There is a mathematical error in the proof.
e.g. 5) Heidi is trying to prove the following number trick:
Choose any number. Add 3. Double it. Add 4. Divide by 2. Take
away the number you started with.
Every time Heidi tries the trick, she gets 5. Her proof, however, does not give the same
result:
Step 1: Choose any number:
n
Step 2: Add 3:
n3
Step 3: Double it:
2n  6
Step 4: Add 4:
2n  10
Step 5: Divide by 2:
2n  5
Step 6: Take away the number you
started with:
n5
Where is Heidi’s error?
Sometimes the error is difficult to spot. See if you can find the error in this proof that 1 = 0.
e.g. 6) Start with a  1 and b  1 .
Step 1:
Then a  b
Step 2:
Multiply both sides by a:
Step 3:
Subtract b 2 from both sides:
Step 4:
Factor each side:
Step 5:
Divide both sides by a  b :
Step 6:
Subtract b from both sides:
Step 7:
Substitute in the original value of a:
What is wrong with this proof?
Assignment:
p. 42-44 #1, 3, 5, 7