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I am trying to determine the truth value of the proposition "If $a^2 = b$ and $b > 0$, then $a=\sqrt{b}$.".

Based on the answer of my teacher, the truth value statement is false.

The counterexample is when $a=-\sqrt{b}$.

My answer is that the truth value of the statement is true.

We know that if $a^2 = b$ and $b > 0$, then $a=\pm\sqrt{b}$. Meaning $a$ can be positive OR negative square root of $b$.

If I will choose only one among the two possible conclusions (positive square root of $b$ OR negative square root of $b$), the statement will still be true.

Please let me know if my understanding is correct.

AYA
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    I hope you'd agree that $(-2)^2=4$. – Angina Seng Jul 11 '20 at 10:42
  • "Meaning $a$ can be positive OR negative square root of $b$" , meaning that $a=\sqrt b$ need not always happen (which is against what the statement says : it says $a = \sqrt b$ always happens), meaning that the statement is false. I do not get how you wrote "If I conclude using any of the possible two conclusions ...", you must explain this statement : in particular, the statement says there is only one possible conclusion, but you have two of them. – Sarvesh Ravichandran Iyer Jul 11 '20 at 10:46
  • @астонвіллатересалисбон Thank you for your explanation. I edit my statement and I hope that I was able to express my idea. What I mean is, I think, that concluding either positive square root of b only or negative square root of b only will still make the statement true. – AYA Jul 11 '20 at 10:52
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    If your statement is to be true then it should be the case that for any $a$ and $b$ I choose, if the hypothesis "$a^2=b$ and $b>0$" is true then the conclusion "$a=\sqrt{b}$" is true. The conclusion does not say "$a=\pm \sqrt{b}$" (that is an entirely different conclusion). So if I choose $a=-1$ and $b=1$ then "$a^2=b$ and $b>0$" is true, but "$a=\sqrt{b}$" is false. – halrankard Jul 11 '20 at 11:41
  • @AnginaSeng. Thank you for your comment. In that case, does the statement "p implies q" is logically equivalent to p always implies q"? – AYA Jul 11 '20 at 11:49

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In mathematics, the meaning of a statement of the form

$$\text{If }A \text{ then }B$$

is

$$\text{If }A \text{ is true, then }B \text{ is } necessarily \text{ true}$$

You have interpreted it as

$$\text{If }A \text{ is true, then }B \text{ }might\text{ be true}$$

which is never a very useful thing to say.

TonyK
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  • Thank you for your answer. I think I have the correct interpretation. I tried to test the truth value. Since for implication, the premise is assumed to be true, and for the statement to be true, the conclusion must be true. Since based on the given, the statement "a is equal to square root of b" is true. Hence, true implies true is true. Therefore the statement is true. – AYA Jul 11 '20 at 12:01
  • My question now is that does the statement "If A, then B." is logically equivalent to If A, then (only) B"? – AYA Jul 11 '20 at 12:06
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    @MAK: "Therefore the statement is true." No, the statement is false. You don't seem to have read my answer. – TonyK Jul 11 '20 at 12:13