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Title: Square root of two is irrational : truth table LOGICAL PROOF
Description: This is a truth table proof of one of the first calculus (real analysis) topics "irrationality of root two". Here I have provided a truth table chart showing that square root of two is irrational (cannot be expressed in p/q form). The usual approach is by the proof by contradiction method. Here we analyze differently with a truth table chart and finally show, even to skeptics who do not believe quickly, that the statement "square root of two is irrational" is indeed true.
Description: This is a truth table proof of one of the first calculus (real analysis) topics "irrationality of root two". Here I have provided a truth table chart showing that square root of two is irrational (cannot be expressed in p/q form). The usual approach is by the proof by contradiction method. Here we analyze differently with a truth table chart and finally show, even to skeptics who do not believe quickly, that the statement "square root of two is irrational" is indeed true.
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This is a truth table proof of one
of the first real analysis topics
"irrationality of root two"
...
The usual approach is
i by the proof by contradiction
Here
we
analyze
n method
...
Page 1
Meaning of symbols:
∀
For all
∃
For some
∈
belongs to
ℕ
Set of all natural numbers
ℙ
Set of all irrational numbers
Page 2
𝟐 irrationality
𝛼, 𝛽 ∈ ℕ
𝑎≝
𝛼
HCF(𝛼, 𝛽)
𝛽
𝑏≝
HCF(𝛼, 𝛽)
⑴
HCF
given
(𝑎 ∈ ℕ)
definitions
(𝑏 ∈ ℕ)
𝛼
𝛽
,
=1
HCF(𝛼, 𝛽) HCF(𝛼, 𝛽)
T
{Theorem}
Page 3
⑵
T
HCF 𝑎, 𝑏 = 1 (𝑎, 𝑏 ∈ ℕ)
{definition of 𝑎 and 𝑏}
⑶
F
CF 𝑎, 𝑏 = 2
{definition of HCF}
⑷
F
𝑎 has factor 2 ^ 𝑏 has factor 2
{definition of CF}
⑸
⑹
⑺
⑻
⑼
𝑎 has factor 2
⟺
𝑎 = 2𝑐 (∃ 𝑐 ∈ ℕ)
⟺
2
𝑎 = 4𝑐 2
⟺
𝑎2 has factor 2
𝑏 has factor 2
⟺
2
𝑏 has factor 2
⟺
2
𝑏 = 2© (∀ © ∈ ℕ)
𝑏2 = 2𝑐 2
⟺
2𝑏2 = 4𝑐 2
2𝑏2 = 𝑎2
⟺
𝑎
2=
𝑏
⟺
𝛼
2=
𝛽
2≠
𝛼
𝛽
T
F
F
{by ⑷ and ⑸(i)}
F
{special case of
⑹(iii)}
F
F
{by ⑸(iii) and ⑺(ii)}
{by ⑸(iv)}
T
{definition of ≠}
Page 4
∀ 𝛼, 𝛽 ∈ ℕ, 2 ≠
⑽
def
𝛼
𝛽
T
{by exact above 9 steps, ⑼ holds ∀ 𝛼, 𝛽 ∈ ℕ}
2∈ℙ
∎
Page 5
Title: Square root of two is irrational : truth table LOGICAL PROOF
Description: This is a truth table proof of one of the first calculus (real analysis) topics "irrationality of root two". Here I have provided a truth table chart showing that square root of two is irrational (cannot be expressed in p/q form). The usual approach is by the proof by contradiction method. Here we analyze differently with a truth table chart and finally show, even to skeptics who do not believe quickly, that the statement "square root of two is irrational" is indeed true.
Description: This is a truth table proof of one of the first calculus (real analysis) topics "irrationality of root two". Here I have provided a truth table chart showing that square root of two is irrational (cannot be expressed in p/q form). The usual approach is by the proof by contradiction method. Here we analyze differently with a truth table chart and finally show, even to skeptics who do not believe quickly, that the statement "square root of two is irrational" is indeed true.