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Proof by Contradiction
1 .Complete the following proof by contradiction to show that √2 is irrational.
𝑎
Assume that √2 is rational. Then √2 can be written in the form of 𝑏, where 𝑎 and 𝑏 are positive
integers without a common factor.
.⋅. 𝑎2 = 2𝑏2
.⋅. 2 is a factor of 𝑎2
.⋅. 2 is a factor of 𝑎, therefore 𝑎 = 2𝑘,
where 𝑘 is an integer
[4]
2. Complete the following proof by contradiction to show that if 𝑛 is a positive integer and
3𝑛 + 2n3 is an odd number, then 𝑛 is an odd number.
We know that 3𝑛 + 2𝑛3 is an odd number.
Assume that 𝑛 is an even number, therefore n= 2𝑘.
[2]
3. Use proof by contradiction to prove the following statement.
When 𝑥 is real and positive,
𝑥+
49
≥ 14
𝑥
The first line of the proof is given below.
Assume that x has a real and positive value so that
𝑥+
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49
< 14
𝑥
4. Complete the following proof by contradiction to show that √3 is irrational.
a
Assume that √3 is rational. Then we can write √3 in the form of b where 𝑎 and 𝑏 are
integers without common factors.
.⋅. 𝑎2 = 3𝑏2 .
.⋅. 3 is a factor of 𝑎2
.⋅. 3 is a factor of 𝑎, therefore 𝑎 = 3𝑘, where 𝑘 is an integer.
5. Complete the following proof by contradiction to show that 𝑥 +
[4]
25
𝑋
≥ 10 where 𝑥 is
real and positive.
Assume that 𝑥 +
25
𝑥
< 10 , where 𝑥
is real and positive.
Because 𝑥 is positive, multiplying both sides of the inequation by 𝑥 gives 𝑥2 + 25 <
10𝑥.
[4]
6. Use proof by contradiction to prove the following statement.
If 𝑎 and 𝑏 are real positive numbers, then 𝑎 + 𝑏 ≥ √𝑎𝑏 .
The first line of the proof is given below
Assume that the real and positive numbers 𝑎 and 𝑏 exist so that 𝑎 + 𝑏 ≤ 2√𝑎𝑏 .
[3]