What does iff really mean and imply? (1 Viewer)

[Giant Lobster]

I know iff == 'if and only if' but Im lost in its technicalities.
When they ask you to prove A is B iff C or similar, whats the difference to asking A is B if C? And also, does iff imply that there is only one way something can occur? e.g. In equalities, if they say a>b iff c = d then does this imply a>b only when c = d and once this is shown adequately, there is no other solutions / cases which will satisfy?
And one more, if they ask you to prove A = B iff c = d, do i have to disprove other cases, e.g. for c != d etc in addition to proving its true for c = d?

 

[wogboy]

When they ask you to prove A is B iff C or similar, whats the difference to asking A is B if C?

A is B iff C implies TWO things:

- If C is true, then A is B.
- If A is B, then C is true

On the other hand, A is B if C implies only ONE thing:

- If C is true, then A is B.
- However there is no guarantee that if A is B, then C is true

And also, does iff imply that there is only one way something can occur? e.g. In equalities, if they say a>b iff c = d then does this imply a>b only when c = d and once this is shown adequately, there is no other solutions / cases which will satisfy?

Yes.

And one more, if they ask you to prove A = B iff c = d, do i have to disprove other cases, e.g. for c != d etc in addition to proving its true for c = d?

Yes you do, unless your working out implicitly disproves the case for c != d.

e.g. prove 2a - 4 = 6b - 4 iff a = 3b

2a - 4 = 6b - 4
iff 2a = 6b (add 4 to both sides)
iff a = 3b (divide both sides by 3)

you have sufficiently proven that 2a - 4 = 6b - 4 iff a = 3b.

Something worth noting is that applying some operations to both sides of an equation does not qualify for an iff such as squaring both sides of an equation, while others do.

e.g. is the following true:
a = b
iff a^2 = b^2 ?

e.g. is the following true:
a + 2 = b + 2
iff a = b

Whenever the case for c != d is not implicitly disproven, you must go a step further and disprove the case for c != d.

e.g. prove that the number n is an odd, single digit composite number iff n=9 (note that composite means not prime)

then you have to show 9 is odd, and composite; but in addition you also have to show that all other single digit numbers 1,2,3,4,5,6,7,8 are not both odd and composite.

 

[Grey Council]

Lobster? Don't you think this guy should be a mod of these forums? or SOMETHING at least? and don't you agree with me that this guy (wogboy) is damn good at maths? and that he is way too damn helpful? (thats a good thing )

 

[kimmeh]

he is doing engineering at uni and its electrical too.. so theres alot of 4 unit maths involved with that

 

[wogboy]

Lobster? Don't you think this guy should be a mod of these forums? or SOMETHING at least? and don't you agree with me that this guy (wogboy) is damn good at maths? and that he is way too damn helpful? (thats a good thing )

lol I think I'll stay an ordinary member for now thanks . Going on these forums gives me something to do during the holidays (and whenever else I have spare time), but I can't promise I'll be here during the uni semesters when the workload gets pretty hectic to say the least.

 

[Giant Lobster]

hmm thanks heaps, I understand now.