A test I took included the question
True or false: SSA triangle problems may have zero or two solutions.
SSA triangles, as was taught in the lesson, can have zero solutions, one solution, or two solutions.
I was unsure if the question asked if zero and two solutions were possibilities with SSA triangles, or if SSA triangles were limited to zero or two solutions. I went with the former (that is, answered True) and got it wrong.
Is it just me, or is the test-question ambiguous?
"Zero or two solutions" suggests to me (I have a maths/physics background) that there can be only zero or two solutions - like a quadratic equation.
The question is not ambiguous.
(True or False) SSA triangles can have zero or two solutions.
The thing with such True or false questions is to first look at the stronger condition. Here the false statement is strong: it rules out 0 and 2 as solutions (cannot be). Comparatively the true statement is vague (can be 0 or 2, can be 1, can be infinity). Use the law of elimination: since the false statement is wrong, true has to be the correct answer.
Finally, as this page shows, an SSA triangle can have 0, 1 or 2 solutions based on the length of the sides and the angles. So mathematically the correct answer for the question is TRUE. (That was one for math.SE.)
As mgb observes, "Zero or two solutions" suggests that there can be only zero or two solutions.
Since there can be zero, one or two solutions, the question is ambiguous - since (as phrased) it implies that one solution is not a viable option (which is false), but states that "zero or two solutions" are viable options (which is true).
There is no ambiguity.
The statement "... can have zero or two solutions" categorically states that it can either have two solutions or none at all. Having one solution or more than two solutions is not a possibility.
The answer to the question must be given accordingly: FALSE.
I find that I could go either way on this one, and much of it depends on whether I start with the question itself, or by examining the question's implied introduction, "True or False?"
Let's start with the question:
SSA triangle problems may have zero or two solutions.
I interpret this as, SSA triangle problems may have zero solutions, or two solutions. So, if an SSA triangle problem has one only one solution, that's not zero, and that's not two. Ergo, if that's possible, then the statement would be FALSE. So, I pick FALSE.
However, now let me start with the T or F? part of the question.
Is this statement TRUE?
SSA triangle problems may have zero or two solutions.
Of course that's true! Some SSA triangle problems have zero solutions, right? (Yes, that's correct.) And others have two solutions, right? (Yes, that's also correct.) Therefore, SSA triangle problems may have zero or two solutions. In other words, I could give you a set of SSA triangle problems (carefully hand-picked, of course) that all have zero or two solutions, and the statement would hold true for that set of SSA triangle problems.
The problem here is that the question was worded ambiguously. Had the question been worded as follows:
All SSA triangle problems have exactly zero or two solutions.
Then clearly the statement would be false. But the problem was obviously not crafted by a skilled logician. The absence of the qualifier "all" and the insertion of the word "may" introduce some very unfortunate ambiguity.
This reminds me of a time where I was playing the game Guess Who with my son. (In the game, you ask a series of Yes-or-No questions that help you narrow down a list of suspects to one specific individual. For example, you might ask, "Does the person have blonde hair?" and a Yes answer would let you eliminate the brunettes, while a No answer would mean the secret individual is blonde. Two players compete, trying to guess the right person first, so it is imperitive to dwindle the list of suspects as quickly as possible.)
Anyhow, my son asked me, "Is the person male or female?" to which I smiled broadly and replied, "Yes."
