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Suppose I have to prove a mathematical statement like

$A$ is $4$ if and only if:

(a) $A = k$

(b) $det(A) = 11$

method 1

do I first prove: $A$ is $4$ if and only if $A = k$

and then prove: $A$ is $4$ if and only if $det(A) = 11$?

method 2

Or do I prove it like: $A$ is $4$ if and only if $A = k$ and $det(A) = 11$

thanks for any answers given

Tortar
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Reuben
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    Depends on the intentions of the author. Usually I would assume method 2 is meant, context notwithstanding, but it's a vague means of writing it (and generally should be avoided). – PrincessEev May 25 '22 at 01:01
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    Of course, an easy diagnostic test is to see if you can find examples where $A=4$ and (a) is not true, or vice versa. (Same can be said about (b) alone.) – PrincessEev May 25 '22 at 01:02
  • Thanks for the answers – Reuben May 25 '22 at 01:04
  • Where have you seen a format such as this? – David K May 25 '22 at 01:38
  • I would expect something like "both of the following hold" or "either of the following hold" after the if and only if. Here, I expect it to be "both of the following hold", because usually I've seen something like "the following statements are equivalent" for the other case. Nonetheless, it's not a good idea to write like this. – ultralegend5385 May 25 '22 at 01:51

1 Answers1

1

Definitely method 2. Consider this goofy example:

  1. A number is equal to 2 if and only if it is a positive integer and it is between 1 and 3.

To prove that statement you start by writing it going forward and in reverse:

  • Going forward: A number is equal to 2 only if it is a positive integer and it is between 1 and 3.

  • Going backward: A number is a positive integer and between 1 and 3 only if that number is 2.

Both of those statements are obviously true! However if you split the original problem similar to your method 1, you will be stuck attempting to prove these:

  1. A number is equal to 2 if and only if it is a positive integer
  2. A number is equal to 2 if and only if it is between 1 and 3

Can you see how each of the statements above are irrevocably wrong? They only work together.

Kartashuvit
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    But the OP does not have "and" between (a) and (b). – mr_e_man May 25 '22 at 02:02
  • What about this: A number is equal to $2$ if and only if it is the smallest positive even integer or it is the square of the diagonal of the unit square. Such statements also can be proved. We don't know which kind the OP encountered because they gave only a nonsense example. – David K May 25 '22 at 02:25
  • @DavidK If we read the OP's given example as P↔(A or B), then their method 1 ((P↔A)∧(P↔B)) is a stronger proof than necessary (the former implies the latter but not vice versa), while their method 2 P↔(A∧B) does not prove/imply the example (P↔(A or B)). I think the OP did mean the example as P↔(A and B), as interpreted in this Answer, and wasn't considering the and/or ambiguity between conditions A and B. – ryang May 25 '22 at 06:33
  • @ryang As noted in comments under the question, the statement to be proved is poorly phrased and ambiguous. As you point out, method 1 is misguided in either case; this doesn't tell us that method 2 is correct, but rather tells us that OP is having difficulty dealing with the problem altogether. To top it off, OP hasn't even given an example of a real problem they've encountered. I think it's better to try to get the OP to actually ask an answerable question first rather than leap to answer a question they might be trying to ask. – David K May 25 '22 at 12:10