The teacher misstated the problem. The problem should have asked find a solution or show that there can be no solution. Or prove or disprove. Now the student

The teacher misstated the problem. The problem should have asked find a solution or show that there can be no solution. Or prove or disprove. Now the student knows that, really, all problems in math are of the form prove or disprove unless the statement is to prove 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. 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 typo. 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". IIRC, Halmos, Finite Dimensional Vector Spaces just states that all the exercises are of the form prove or disprove. 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". 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. 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 prove or disprove.