I have worked out the problem A B C, my question is in the last statement " If the numbers e, π, π^2, e^2 and eπ are irrational, prove that at most one of the numbers π+e, π−e, π^2−e^2, π^2+e^2 is rational." can i prove the contrapositive of it (i.e.at least two of the numbers π+e, π−e, π^2−e^2, π^2+e^2 is rational, then at least one of the e, π, π^2, e^2 and eπ are rational) to be true to prove the statement?
What does it mean to say that a number x is irrational? Prove by contradiction statements A and B below, where p and q are real numbers.
A: If pq is irrational, then at least one of p and q is irrational.
B: If p + q is irrational, then at least one of p and q is irrational.
Disprove by means of a counterexample statement C below, where p and q are real numbers.
C: If p and q are irrational, then p + q is irrational.
If the numbers e, π, π^2, e^2 and eπ are irrational, prove that at most one of the numbers π+e, π−e, π^2−e^2, π^2+e^2 is rational.
Yes, proving the contrapositive also proves the statement.
We need to be sure the contrapositive is really a contrapositive. "At least two" is the exact negation of "at most one," and "at least one is rational" is the exact negation of "all are irrational," so I think you're doing OK so far.