Use contradiction method to prove that :“p: √3 is irrational” is

1.	Use contradiction method to prove that :“p: √3 is irrational” is a true statement.
Answer» Contradiction statement: √3 is a rational number. Proof: If √3 is a rational number, then √3 = p/q where (p, q) co-prime. or q = p/√3 or q² = p^2/3 ….(1) Thus, p² must be divisible by 3. Hence p will also be divisible by 3. We can write p = 3k, where k is a constant. ⇒ p² = 9c² (1)⇒ q² = 9c^2/3 = 3c² or c² = q^2/3 Thus, q² must be divisible by 3, which implies that q will also be divisible by 3. Thus, both p and q are divisible by 3. Which is a contradiction, as we assume that p and q are co-prime. Thus, √3 is irrational. Hence, the statement p is true.

Use contradiction method to prove that :“p: √3 is irrational” is a true statement.

Answer»

Contradiction statement: √3 is a rational number.

Proof:

If √3 is a rational number, then √3 = p/q where (p, q) co-prime.

or q = p/√3

or q² = p^2/3 ….(1)

Thus, p² must be divisible by 3. Hence p will also be divisible by 3.

We can write p = 3k, where k is a constant.

⇒ p² = 9c²

(1)⇒

q² = 9c^2/3 = 3c²

or c² = q^2/3

Thus, q² must be divisible by 3, which implies that q will also be divisible by 3.

Thus, both p and q are divisible by 3.

Which is a contradiction, as we assume that p and q are co-prime.

Thus, √3 is irrational.

Hence, the statement p is true.