1.5 PROOFS THAT ARE NOT VALID
Invalid Proof: A proof (or argument) that contains an error in reasoning or that
contains an invalid assumption or generalization.
Examples:
1) All musicians have long hair. Mr. Rawlyk is a musician. Mr. Rawlyk has long hair.
2) Romney claims that he can prove that -3 = 3
Romney’s Proof:
STEP 1: Assume that -3 = 3
STEP 2: Square both sides (-3)2= (3)2
STEP 3: This gives a true statement: 9 = 9
STEP 4: This means the assumption -3 = 3 is correct
3) Cher created a math trick in which she always ended with 4 as an answer. Look
at her proof below and explain where the mistake is and indicate if it is an invalid
assumption or an error in reasoning.
Cher’s Proof:
STEP 1:
STEP 2:
STEP 3:
STEP 4:
STEP 5:
Let n = any #
Multiply by 2
Add 8
Divide by 2
Subtract original #
n
2n
2n + 8
2n + 4
n+4
4) Proof 5 = -5
If we assume: 5 = -5
Squaring both sides: 25 = 25
Square root both sides: + 5 = + 5
What is the error?
5) Suppose:
a+b=c
This can be rewritten as:
4a – 3a + 4b – 3b = 4c – 3c
Reorganize:
4a + 4b – 4c = 3a + 3b – 3c
Factor:
4(a + b – c) = 3(a + b – c)
Divide both sides by (a + b – c):
4=3
Where is the error in deduction?
Key Ideas
* A ________ error in reasoning can break down the logical argument of a proof.
* This will result in an __________conclusion.
* Division by _______ always creates an error in a proof.
* ____________ reasoning must be avoided. Be careful not to assume a result
that follows from what you are trying to prove.
1.5 PROOFS THAT ARE NOT VALID
Invalid Proof: A proof (or argument) that contains an error in reasoning or that
contains an invalid assumption or generalization.
Examples:
1) All musicians have long hair. Mr. Rawlyk is a musician. Mr. Rawlyk has long hair.
ANSWER: There is an invalid assumption “All musicians have long hair”
which makes the entire argument not valid.
2) Romney claims that he can prove that -3 = 3
Romney’s Proof:
STEP 1: Assume that -3 = 3
STEP 2: Square both sides (-3)2= (3)2
STEP 3: This gives a true statement: 9 = 9
STEP 4: This means the assumption -3 = 3 is correct
ANSWER: There is an invalid assumption in STEP 1. -3 ≠ 3
3) Cher created a math trick in which she always ended with 4 as an answer. Look
at her proof below and explain where the mistake is and indicate if it is an invalid
assumption or an error in reasoning.
Cher’s Proof:
STEP 1: Let n = any #
n
STEP 2: Multiply by 2
2n
STEP 3: Add 8
2n + 8
STEP 4: Divide by 2
2n + 4
STEP 5: Subtract original # n + 4
ANSWER: There is an error in reasoning in STEP 4. When you divide
2n + 8 by 2 you get n + 4
Corrected Proof:
STEP 1: Let n = any #
n
STEP 2:
Multiply by 2
2n
STEP 3: Add 8
2n + 8
STEP 4: Divide by 2
n + 4
←
STEP 5: Subtract original #
4
←
4) Proof 5 = -5
If we assume: 5 = -5
Squaring both sides: 25 = 25
Square root both sides: + 5 = + 5
What is the error?
5 = -5 started with a false assumption (circular reasoning)
5) Suppose:
a+b=c
This can be rewritten as:
4a – 3a + 4b – 3b = 4c – 3c
Reorganize:
4a + 4b – 4c = 3a + 3b – 3c
Factor:
4(a + b – c) = 3(a + b – c)
Divide both sides by (a + b – c):
4=3
Where is the error in deduction?
cannot divide by zero - third step
Key Ideas
* A single error in reasoning can break down the logical argument of a proof.
* This will result in an invalid conclusion.
* Division by zero always creates an error in a proof.
* Circular reasoning must be avoided. Be careful not to assume a result that
follows from what you are trying to prove.