In the following argument to show that √2 is irrational, give proofs appropriate for steps 3, 5 and 6.

1. Assume that √2 is rational.

2. Define the set S to be the set of positive integers with the following property:

n is in S if and only if n√2 is an integer.

3. Show that the set S contains at least one positive integer.

4. Define the integer k to be the smallest positive integer in S

5. Show that (√2-1)k is in S.

6. Show that steps 4 and 5 are contradictory and hence that √2 is irrational.

Prove that is rational if and only if is rational.

Use an argument similar to that of part (i) to prove that and are irrational.

Click here for solution by Yuliia Tereshchuk

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