The teacher misstated the problem. The problem should have asked <i>find a solution or show that there can be no solution</i>. Or <i>prove or disprove</i>.<p>Now the student knows that, really, all problems in math are of the form <i>prove or disprove</i> unless the statement is to <i>prove</i> in which case it is misleading to have the claim false. But, in texts, there can be errors, and a student needs to know that.<p>In college, I was reading a book on group theory and could not confirm a statement in the book. Some hours went by, and I couldn't get it. Eventually we found a counterexample and concluded that the book had an error. Actually, it was just an error in typography, a <i>typo</i>.<p>Later, on a Ph.D. qualifying exam, I struggled too long with a problem and got a failing grade. Yup, the problem asked for a proof, but the claim was false. There was a typo. I appealed, got an oral makeup exam in front of several profs, some angry, and ended with a "High Pass".<p>IIRC, Halmos, <i>Finite Dimensional Vector Spaces</i> just states that all the exercises are of the form <i>prove or disprove</i>. He goes on to say, "then discuss such changes in the hypotheses and/or conclusions that will make the true ones false and the false ones true".<p>At one point one course, for some early homework, my submission was that nearly all the exercises were false -- I'd found that the claims failed on the empty set! Given that the course started out with such sloppy work, I dropped it.<p>Net, really, in general in practice in math, all the statements, due to errors, typos, or whatever, have to be regarded as of the form <i>prove or disprove</i>.