Question

Discrete. I am trying to get better at proofs, especially the notation. A product of any two irrational numbers is
irrational.

Answer 1

So I would like to let statement r1=(Let m = sqrt(2) and Let n = sqrt(2)

Answer 2

Therefore r1 implies r2 with contradicts our original statement. So The initial statement " A product of any two irrationals," is False.

Answer 3

Going for proof by contradiction...

Answer 4

that's one counter example

another is let m = sqrt(3) and n = sqrt(12)

m & n are definitely irrational, but m*n = sqrt(3)*sqrt(12) = sqrt(3*12) = sqrt(36) = 6

which shows that m*n is rational

Answer 5

So should I try to prove by cases and show multiple? Or is there a way I can just do one example for proof by contradiction?

Answer 6

once you show one counterexample, it disproves the whole statement

Answer 7

because the claim is for ALL irrational numbers (or any two irrationals)

Answer 8

find one hole and the whole thing falls apart

Answer 9

Thank you Jim!