Proof by contradiction

Question 1

What is the first step in a proof by contradiction?

Accepted Answer

Assume the statement is not true

Question 2

What statement are we proving with proof by contradiction?

Accepted Answer

“If x² is even, then x must be even”

Question 3

What happens if we assume the statement is false?

Accepted Answer

There must be an odd number for x for which x² is even

Question 4

How can an odd number x be expressed?

Accepted Answer

as x = 2k + 1, where k is an integer

Question 5

What is the expression for x² when x is odd?

Accepted Answer

x² = (2k + 1)² = 4k² + 4k + 1

Question 6

What does the expression 4k² + 4k represent?

Accepted Answer

it is even because it is 2 times an integer

Question 7

What conclusion do we reach if x² is even?

Accepted Answer

It contradicts the assumption that there is an odd number x for which x² is even

Question 8

What can be concluded about x if x² is even?

Accepted Answer

x must be even, hence the original statement is true

Question 9

What is the assumption made to prove that surds are irrational?

Accepted Answer

Assume that √2 can be written as a with a and b both non-zero integers

Question 10

What assumption is made about a and b in the proof involving √2?

Accepted Answer

a and b do not have any common factors

Question 11

What does the equation √2 = a imply?

Accepted Answer

√2b = a

Question 12

What do we get when we square both sides of √2 = a?

Accepted Answer

2b² = a²

Question 13

What can be concluded about a² in the proof regarding √2?

Accepted Answer

a² is an even number

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