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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?

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5  
Since Shyam and mgb come to different conclusions, each for good reasons, I suspect the question is ambiguous. –  Henry May 3 '12 at 6:21
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Yeah, I'd say you got this right, if you use the strictly logical definition of OR. SSA problems may have zero solutions - true. SSA problems may have one solution - true. SSA problems may have two solutions - true. Strictly speaking, "SSA problems may have zero or two solutions" means the same as "SSA problems may have zero solutions OR they may have two solutions". True OR True gives True; even though there is a third possibility. However, the fact that so many people have answered this question the opposite way from me seems to indicate (as Henry says) that it's truly ambiguous. –  user16269 May 3 '12 at 7:05
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Eyes can have the colour green or blue — True. Eyes can only have the colour green or blue — False. –  donothingsuccessfully May 3 '12 at 7:11
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As @donothingsuccessfully indicates, the addition of only removes the ambiguity (and makes the answer False). Or rather, it changes the meaning entirely. Because the original question does not have only, it could easily be interpreted as saying "among other numbers of solutions" and the answer is True. –  Andrew Leach May 3 '12 at 7:40

5 Answers 5

up vote 4 down vote accepted

"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.

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Ha, nice XKCD avatar. Can't it be interpreted the other way though? Wouldn't putting false say that SSA triangles may not have zero or two solutions, which is wrong because they can? –  mowwwalker May 3 '12 at 5:01
    
Quadratic equation: zero or two real solutions (I think!) –  user19148 May 3 '12 at 5:02
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Huh? x^2 - 6x + 9 = 0 has only one solution (x=3). –  user16269 May 3 '12 at 7:00
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And what's a quad eqn with zero solution? –  Bravo May 3 '12 at 7:07
    
@Shyam: x^2 - 6x + 10 = 0 has no real solutions –  Henry May 3 '12 at 7:16

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."

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The question is not ambiguous.

(True or False) SSA triangles can have zero or two solutions.

  • If your answer is true, then you mean the cardinality of the solution set can be zero or two. There is a possibility that either of these could be the value. It's equally possible that neither is.
  • If your answer is false, then neither 0 nor 2 can ever be the cardinality. To give another example, consider (True or false) The number of seeds in a monocotyledon can be two. It is false, as the number is always one.

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.)

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Thanks for editing the q, @jwpat7. Will tune my answer now. Hopefully your interpretation is correct. –  Bravo May 3 '12 at 5:20
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(1) Why mention the SAS case? How does it have any bearing? The whole last paragraph of your answer is a set of non sequiturs. (2) What does "If the answer is a true" mean? Also, please put the bullets into parallel form, rather than starting one with "If the answer..." and the other with "The answer is..." (3) The OP's question is not about the math, it is about whether the test question is ambiguous. –  jwpat7 May 3 '12 at 5:30
    
@jwpat7: I considered "true" to be the shortened form of "the answer TRUE" and hence used "a true". For your 3rd q, adding the Wiki link will give the context for our non-math friends here. More info never hurts. –  Bravo May 3 '12 at 5:36
    
Phrase "the answer" as you are using it in your answer is ambiguous, because of not clearly distinguishing among the correct answer to the question, the test-giver's answer to the question, the test-taker's answer, etc. You could instead say "If one answers True, they mean that ...", and so forth. –  jwpat7 May 3 '12 at 5:49

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.

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1  
But may is only a possibility; doesn't rule out the possibility of 1 solution, does it? –  Bravo May 3 '12 at 7:05
    
may does not imply a possibility. Here it is used as in "You may come now." (You are permitted to come now.) and "You may not enter after 10:00 am." (You are not permitted to enter after 10:00 am.) –  Kris May 3 '12 at 8:01
    
@Kris: Sometimes 'may' means a possibility, sometimes allowance. –  Mitch May 3 '12 at 15:20
    
In the context, it means that only zero and two are legal (technically) or permissible. cf. An integer variable may only be assigned integer values. (Of course, this is not true.) –  Kris May 3 '12 at 15:35

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).

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