Question

(8 pts) t by Contradiction and by (4 pts) 4. Given the statement, V real numbers x, if x2 is irrational then x is irrational. Write what you would suppose and what you need to show to prove this statemen Contraposition. Don't write a complete proof. a. By Contradiction (4 pts) b. By Contraposition

Answer 1

Proof By Contradiction:

Let x be a real number. We want to prove by contradiction that if x² is irrational, then x is irrational.

Proof: Assume the negation of this statement: x² is irrational and x is rational.

But if x is rational, then x can be written as x=b/c where b and c are integers.

Then x²=b²/c² which is also rational.

This contradicts the statement that x² is irrational.

Thus, if if x² is irrational, then x is irrational.

proof by contraposition:

we need to prove a contrary statement.. which is

if x is rational then x² is rational..

now.. x is rational so we can write it in x = a/b where a and b are two integers..

now squaring both sides..

x² = a² / b²

we can clearly see than which is also in rational form..

so we have proved that.. if x is rational then x² is rational.. which.. will proof the original statement by contraposition..