If and only if

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  • #1
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Main Question or Discussion Point

Is it bad practice to use ⇒ in places where ⇔ is more appropriate?
In textbooks I often see things like:

##3x + 2 = 6##
##⇒ 3x = 4##
##⇒x = \frac{4}{3}##

Isn't the us of "if" here technically wrong, since the reverse statements are also implied?
 

Answers and Replies

  • #2
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It isn't wrong because [itex] (A \Leftrightarrow B) \Rightarrow (A\Rightarrow B) [/itex].
 
  • #3
Stephen Tashi
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It isn't bad practice to use [itex] \implies [/itex] in place of [itex] \iff [/itex] when the argument only needs [itex] \implies [/itex]. If the argument is trying to "work backwards" from an assumption to a true statement and then leave it to the reader to reverse all the implications to make a real proof then [itex] \iff [/itex] should be used.

It is a cultural tradition in writing mathematics that one may use "if" to mean "if and only if" when making definitions. For example, a book might say "We will say that an integer k is "even" if it k/2 is an integer". Strictly speaking that definition doesn't rule-out 3 as being an even integer. It merely fails to comment on whether 3 is even. However, tradition says that you interpret "if k/2 is an integer" to be "if and only if k/2 is an integer". It's better practice (in my opinion) to use "iff" as an abbreviation for "if and only if" when writing definitions that intend to convey "if and only if".
 
Last edited:
  • #4
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Is it bad practice to use ⇒ in places where ⇔ is more appropriate?
In textbooks I often see things like:

##3x + 2 = 6##
##⇒ 3x = 4##
##⇒x = \frac{4}{3}##

Isn't the us of "if" here technically wrong, since the reverse statements are also implied?
As already mentioned by Shyan and Stephen, it isn't wrong to use ⇒ here. All three of your equations above are equivalent, which means that they all have exactly the same solution set.

From the perspective of solution sets, if the solution set of one equation is a subset of another equation, the first equation "implies" the second.

A simple example can shed some light.
x = 2
⇒ x2 = 4
The solution set of the first equation is {2}. The solution set of the second equation is {2, -2}.

Unlike the equations in your example, these two equations are not equivalent, as they have different solution sets, so the following implication is incorrect.
x2 = 4
⇒ x = 2.
 
  • #5
479
20
As already mentioned by Shyan and Stephen, it isn't wrong to use ⇒ here. All three of your equations above are equivalent, which means that they all have exactly the same solution set.

From the perspective of solution sets, if the solution set of one equation is a subset of another equation, the first equation "implies" the second.

A simple example can shed some light.
x = 2
⇒ x2 = 4
The solution set of the first equation is {2}. The solution set of the second equation is {2, -2}.

Unlike the equations in your example, these two equations are not equivalent, as they have different solution sets, so the following implication is incorrect.
x2 = 4
⇒ x = 2.
A correct implocation would be ##x^2 = 4 ⇒ x = \pm 2##, right?
 
  • #6
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A correct implocation would be ##x^2 = 4 ⇒ x = \pm 2##, right?
Yes. And since both equations are equivalent, you could use ⇔ between them.
 
  • #7
FactChecker
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No. It is more appropriate to only use the implication that is needed for your proof. Otherwise, every statement has unnecessary complications, distractions, and possible errors. It would not be clear to the reader what is essential to the proof and what is not.
 
  • #8
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No. It is more appropriate to only use the implication that is needed for your proof.
What (and who) are you disagreeing with. My comments were only in the context of the simple example I gave, that contrasted the difference between, for lack of better terms, a one-directional implication and a bi-directional implication. It was not intended to mean that the bi-directional implication (##\Leftrightarrow##) should be used all the time.
FactChecker said:
Otherwise, every statement has unnecessary complications, distractions, and possible errors. It would not be clear to the reader what is essential to the proof and what is not.
 
  • #9
FactChecker
Science Advisor
Gold Member
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What (and who) are you disagreeing with..
Well, I was referring to the OP.
 
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