I love to distribute, maybe a little too much
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4 solutions
The problem with this question lies not in the answers that are being posted, but with the wording of the question. The Question states "True or False ... for all real numbers a and b. The actual situation, has two subsets of real numbers, that need to be considered. If either a or b is zero, then the statement is true. If both a and b are both non-zero, then the statement is false. Thus the writer cannot ask "True or False". This is sloppy wording, and as an professional scientist and mathematician for over 50 years, and a technical editor the question has to be corrected. One also has to consider that if this question was given on an examination where no multiple choices answers were provided, then the answer would clearly have to state the neither answer "True or False" is valid and show that two sets of real numbers exist, one that satisfies the equation and the other that does not.
I did the same
To expand on James Clemens' response, the answer "It's true if and only if a=0 and/or b=0" is a correct statement. A subset of that group would be if a or b is a negative number in which case the equation then becomes false. However, as explained by the author's solution, the equation can only be true when (If and only if) the two conditions are met, one of which is that a=0 and/or b=0. No where in the answer does it indicate that the equation is ALWAYS true when a=0 and/or b=0.
In other words, the equation is true if and only if a=0 and/or b=0, however other other conditions need to be met as well.
I agree that this is a very poorly worded question.
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No, the statement “it’s true only if a=0 and/or b=0” would be correct. However, the statement “it’s true if (and only if) a=0 and/or b=0” is not a correct statement. It is false in the same way that the statement “it’s true if a=0 and/or b=0” is not a correct statement. ‘If AND only if’ is not the same as ‘only if’.
Yeah, I said option B is correct because it's true for a=0 AND b=0
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I agree. If a=0 then we are left with b and the square root of b2 is b! The same applies to when b=0.
So, basically those of us who chose the second answer are incorrect because it is not enough to specify that a = 0 and/or b = 0. We must ALSO remember to qualify that with: if a = 0, then b ≥ 0 and if if b = 0, then a ≥ 0.
Ugh, I didn't even consider the negatives. My System 2 is lazy today...
I didn't get the answer right (stupid me), but I believe that "none" is the subject of the sentence "None of the other options are correct." As "none" is a singular noun, should the sentence not read "None of the other options is correct."? Isn't "of the other options" an adjectival clause, not the subject of the sentence? Please, could a better grammarian than I, correct me if I am wrong?
I think that the problem is rather that the metalanguage (i.e. English) is poorly defined.
How do I have to read "and/or"?
I was always taught, the correct answer (using propositional logics) is the "or"-notation (symbol "v"). [Inclusive, meaning that either a=0 or b=0 or both are equal to 0.]
So, "a=0 v b=0" is what I'd have expected to be the answer...
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What most people seem to be misunderstanding is actually the ‘if and only if’. There seems to be a conflation with one of either ‘if’ or ‘only if’ instead of it being read as ‘if AND only if’ from what I have read of the comments. The second answer actually consists of two statements: an ‘if’ statement and an ‘only if’ statement. Whilst the ‘only if’ statement is true, the ‘if’ statement is not, and consequently the statement as a whole is false.
Easy to testify: Ask ANY computer/AI in the world the very same question. It will keep "None of the other options are correct" for later to check the 'others' before, iterate throught the alternatives, mark TRUE for the "It's true if and only if a=0 and/or b=0", (because it IS true indeed, despite not being complete for the true requirement), check the "None of the other options are correct" FALSE of course because one of them is already TRUE, and end of story. If a code is not returning what you expected, your logic is wrong. Period. BAD question.
In the link to "Is x 2 = ± x ?", it says that, and I quote: " x is defined as 'the non-negative real number which, when squared, equals x.' " Key word: defined. It's somewhat of an arbitrary definition. Of course, the page in question also says that " x is defined this way so that it is a function " . But that still doesn't satisfy: why would x HAVE to be a (single-valued) function? Why couldn't it be a Multi-Valued Function?
NOWHERE in the question asked above is it stated that we are talking about the "positive square root" or the "single-valued square root". To demand those answering the question to follow a single, arbitrary definition of x -- in other words, to check the Wiki pages before giving an answer just in case Brilliant follows an arbitrary definition with which they are either not familiar or which differs from their own definition -- is more about having others fall in line with your own preferences than it is about teaching proper Mathematics.
Of course, there are plenty Mathematicians who also insist on defining x as + x instead of simply (and far more naturally) the inverse of x 2 , or in other words ± x . This doesn't mean that such a definition is any less arbitrary and biased: it just means that Mathematicians, too (like most Human Beings, actually), have a tendency to push their own preferences unto others as either objective or logical fact, when in reality they're nothing of the sort.
Your first link has changed in to x 2 y 2 = ( x y ) 2
As mentioned before, the logical expectations of this question do not align. The 2nd answer is true, and hence nullify the last answer (because it itself is actually false). So why that is the correct answer, is beyond me...
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The second answer is not true. ‘If and only if’ is a stronger statement than ‘only if’.
Consider the statements: ‘I have milk in my coffee if (and only if) it is before 10am’ versus ‘I have milk in my coffee only if it is before 10am’.
In the first statement, I always have milk in my coffee if it is before 10am. However, in the second statement, I can still have coffee without milk before 10am, since the statement only elaborates on what I cannot do after 10am.
Counterexample: Suppose a = 0 and b = − 2 . Then, a 2 + b 2 = 0 + ( − 2 ) 2 = 2 . (Since a 2 = ∣ a ∣ .) However, a + b = 0 + ( − 2 ) = − 2 .
This shows there are some real-valued a , b for which the equality is false, even when " a = 0 and/or b = 0 ". Because the equality is true for a = 0 and non-negative b , it is not false for all real numbers. Hence, the only valid answer is "None of the other options are correct".
It should be that the second answer is correct provided that either a or b does not have a negative notation when the other is zero..
I think 2nd answer is correct too. Consider that a=0 and b=2 if we sqare them 0+4=4 and then take the root of those thats 2. Now a+b is going to be equal 2 cus 0+2 but this works only if the either a or b have a postive valueor if a or b equal zero.
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The second answer says "if and only if". For the answer to be true, then two things must hold. For all real a , b :
- If the expressions are equal, then a = 0 and/or b = 0 .
- If a = 0 and/or b = 0 , then the expressions are equal.
The first is true, but the second is false (e.g. a = 0 , b = − 2 ). So, the second answer is not correct.
But ((-5)^(1/2))^2 is - 5 in complex notation..
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The problem says that a and b are real numbers. So, − 5 is undefined.
“None of the other options are correct” is incorrect!
It's kind of a sleezy question. Answer 2 is correct when the non zero component is positive and wrong when it is negative. I am out of here.
The a and b need to be in brackets with or without the square, otherwise it is very ambiguous
The correct choice is “None of the above answers IS correct”. (as the word NONE is singular not plural)
Only 15% success on this question.
Note that the case where A and B both equal 1 is true. But not for -1. The case for A and or B = 0 is OK.
But, for example A = 2 and B =3 4+9=13. square root of 13 isn't 5.
So I am a bit stupid when it comes to math but I had recently learned in my class that if you have the squared root of something squared it cancels out the squares? I tried to apply it here which would put the answer at true which is not correct but im pretty sure all of this is way over my head anyway it was an attempt I suppose
The exceptional cases when the proposed identity is true are those that meet both of these conditions:
For example, the equation is true when a = 5 and b = 0 , and when a = b = 0 .
And the equation is false when a = 5 and b = 5 and when a = − 2 , and b = 0 .
Therefore, the correct answer from the given options is: "None of the other options are correct."
Proof:
Consider squaring both sides of the suggested identity.
(Equivalency should hold, since if a = b , it would also be the case that a 2 = b 2 ):
Left Hand Side: ( a 2 + b 2 ) 2 = ∣ a 2 + b 2 ∣ = a 2 + b 2
(The last step is valid because a 2 and b 2 must both be non-negative.)
Right Hand Side: ( a + b ) 2 = a 2 + b 2 + 2 a b
Comparing the two sides: a 2 + b 2 and a 2 + b 2 + 2 a b , it is clear that the proposed identity is only true in the cases where 2 a b = 0 . This will only occur if either a = 0 or b = 0 . Additionally, any non-zero variable must be positive, such that a 2 + 0 = ∣ a + 0 ∣ = a , or similarly for b . In all other situations, the equation is false.
For more details, see the following two misconceptions pages: