Question Type 2: Using contradiction without conditions Exercises

Prove that $\sqrt{p}$ is irrational for any prime $p$.

Prove that $\sqrt{2}$ is irrational.

Prove that there are infinitely many primes.

Prove that the cube root of 2 is irrational.

Prove that $\sqrt{2}+\sqrt{3}$ is irrational.

Prove that there is no largest even integer.

Prove that the sum of a rational number and an irrational number is irrational.

Prove that between any two distinct real numbers there exists an irrational number.

Prove that $\log_{10} 2$ is irrational.

Prove that the set of real numbers in $[0,1]$ is uncountable.

Prove that $\sqrt{3}$ is irrational. [6 marks]

Prove that if an integer $a$ is such that $a^2$ is even, then $a$ is even.