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    I just muddled myself up with some implications. Let's take x^2 - x = x(x-1) = 0 to be our quadratic equation.

    Am I right in saying that x=1 \implies x(x-1) = 0 but not that x(x-1) = 0 \implies x=1 because it could be the case that x=0.

    Is the above correct? If so - can you explain it a little more clearly? Perhaps with a different logic example to cement my understanding?

    I can however state that x(x-1) = 0 \iff (x=0 \text{ or } 1) - right?
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    (Original post by Zacken)
    I just muddled myself up with some implications. Let's take x^2 - x = x(x-1) = 0 to be our quadratic equation.

    Am I right in saying that x=1 \implies x(x-1) = 0 but not that x(x-1) = 0 \implies x=1 because it could be the case that x=0.
    Yes.

    Is the above correct? If so - can you explain it a little more clearly? Perhaps with a different logic example to cement my understanding?

    I can however state that x(x-1) = 0 \iff (x=0 \text{ or } 1) - right?
    Yes.

    May help, or may be too simplistic.

    ab = 0 implies a=0 or b=0, including the possibility that both could, depending on circumstance.

    If c = 0 then c times "anything" = 0.

    Edit:
    May help with your example (sorry!)

    x=0 implies x(x-1) =0

    x=1 implies x-1=0 implies x(x-1) = 0

    Then via the rules of logic (or-inclusion)

    (x=0 OR x=1) implies x(x-1) =0

    Edit2!

    Don't know of I hit the bit that's caused the confusion there. Bit of carpet bombing.
    Apologies for everything that's totally obvious, if not all of it.
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    (Original post by ghostwalker)
    Yes.



    Yes.

    May help, or may be too simplistic.

    ab = 0 implies a=0 or b=0, including the possibility that both could, depending on circumstance.

    If c = 0 then c times "anything" = 0.

    Edit:
    May help with your example (sorry!)

    x=0 implies x(x-1) =0

    x=1 implies x-1=0 implies x(x-1) = 0

    Then via the rules of logic (or-inclusion)

    (x=0 OR x=1) implies x(x-1) =0

    Edit2!

    Don't know of I hit the bit that's caused the confusion there. Bit of carpet bombing.
    Apologies for everything that's totally obvious, if not all of it.
    My OP was probably unclear, your post was still valuable in clearing up some confusion and giving me an example! And I think it kinda helped me click onto something that made me understand what I was asking, can you just verify if the below is correct?

    I'm more inquiring as to the direction of implications and when something is an iff.

    My understanding is that a iff b is:
    if a then b
    if b then a.

    So, for example: x(x-1) = 0 iff (x=0 or x=1) because
    if x(x-1) = 0 then (x=0 or x=1)
    if (x=0 or x=1) then x(x-1) = 0.

    If the above is correct, then I've just got to clear up my understanding of sufficient and necessary conditions.

    I think I've grasped the fact that a \iff b means that a is necessary and sufficient for b and b is necessary and sufficient for a. (I may have messed the order of words there. :lol:)

    So (x=0 or x=1) is necessary and sufficient for x(x-1) = 0
    and x(x-1) = 0 is necessary and sufficient for (x=0 or x=1).

    But I'm not sure about what only necessary or only sufficient means in the sense of implication directions.


    Would x=1 be sufficient but not necessary for x(x-1) = 0?

    Edit: Oh, and PRSOM.
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    (Original post by Zacken)
    My OP was probably unclear, your post was still valuable in clearing up some confusion and giving me an example! And I think it kinda helped me click onto something that made me understand what I was asking, can you just verify if the below is correct?

    I'm more inquiring as to the direction of implications and when something is an iff.

    My understanding is that a iff b is:
    (if a then b
    AND
    if b then a).
    Yes - added the AND and parentheses to avoid any confusion.

    So, for example: x(x-1) = 0 iff (x=0 or x=1) because
    if x(x-1) = 0 then (x=0 or x=1)
    if (x=0 or x=1) then x(x-1) = 0.

    If the above is correct, then I've just got to clear up my understanding of sufficient and necessary conditions.
    Yes.

    I think I've grasped the fact that a \iff b means that a is necessary and sufficient for b and b is necessary and sufficient for a. (I may have messed the order of words there. :lol:)
    Note:
    "a is neccessary and sufficient for b" means the same as "b is necessary and sufficient for a". Here's a wiki exposition on it.

    So (x=0 or x=1) is necessary and sufficient for x(x-1) = 0
    and x(x-1) = 0 is necessary and sufficient for (x=0 or x=1).

    But I'm not sure about what only necessary or only sufficient means in the sense of implication directions.


    Would x=1 be sufficient but not necessary for x(x-1) = 0?
    Yes.

    "x is an integer" is necessary by not sufficient.
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    (Original post by Zacken)
    I'm more inquiring as to the direction of implications and when something is an iff.

    My understanding is that a iff b is:
    if a then b
    if b then a.
    Yup that's it.

    (Original post by Zacken)
    So, for example: x(x-1) = 0 iff (x=0 or x=1) because
    if x(x-1) = 0 then (x=0 or x=1)
    if (x=0 or x=1) then x(x-1) = 0.

    If the above is correct, then I've just got to clear up my understanding of sufficient and necessary conditions.

    So (x=0 or x=1) is necessary and sufficient for x(x-1) = 0
    and x(x-1) = 0 is necessary and sufficient for (x=0 or x=1).

    But I'm not sure about what only necessary or only sufficient means in the sense of implication directions.


    Would x=1 be sufficient but not necessary for x(x-1) = 0?

    Edit: Oh, and PRSOM.
    Yes it would be sufficient but not necessary.

    Another good example of a sufficient but not necessary condition is that being divisible by 6 say, is sufficient but not necessary for a number being divisible by 3.
    On the other hand it is necessary, but not sufficient for a number to be divisible by 3 if it is divisible by 6.
    You often need to be careful with logic when proving things in a way that isn't emphasised at A-level or below, because obviously false statements can imply true statements.

    There are some other nice things you might like to look into, such as the contrapositive.
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    (Original post by Zacken)
    I just muddled myself up with some implications. Let's take x^2 - x = x(x-1) = 0 to be our quadratic equation.

    Am I right in saying that x=1 \implies x(x-1) = 0 but not that x(x-1) = 0 \implies x=1 because it could be the case that x=0.

    Is the above correct? If so - can you explain it a little more clearly? Perhaps with a different logic example to cement my understanding?

    I can however state that x(x-1) = 0 \iff (x=0 \text{ or } 1) - right?
    I would write your final example as x(x-1) = 0 \iff (x=0 \text{ or } x=1)

    However, you can split this up into the two equivalent statements:

    x(x-1) = 0 \Rightarrow (x=0 \text{ or } x=1)
    (x=0 \text{ or } x=1) \Rightarrow x(x-1)=0

    You can read the first as:

    For all x such that x(x-1) = 0 it is true that (x=0 \text{ or } x=1)

    This is true since if x=0 then the statement is:

    (0(0-1)=0 \Rightarrow (0=0 \text{ or } 1=0)) \Rightarrow (\text{T} \Rightarrow \text{T} \text{ or } \text{F}) \Rightarrow (\text{T} \Rightarrow \text{T}) \Rightarrow (\text{T})

    where T means true and F means false and I've applied the rules of Boolean logic to show that the first statement implies "true", and is therefore true itself. The same approach works if x=1

    However if we try this with x(x-1) = 0 \implies x=1, then we find that the equivalent statement is:

    For all x such that x(x-1) = 0 it is true that x=1

    But this is false since if x=0 we have the logical equivalence:

    (0(0-1)=0 \Rightarrow 1=0) \Rightarrow (\text{T} \Rightarrow \text{F}) \Rightarrow (\text{F})

    so we have a statement implying "false", so that statement is false.
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    (Original post by atsruser)
    However if we try this with x(x-1) = 0 \implies x=1, then we find that the equivalent statement is:

    For all x such that x(x-1) = 0 it is true that x=1

    But this is false since if x=0 we have the logical equivalence:

    (0(0-1)=0 \Rightarrow 1=0) \Rightarrow (\text{T} \Rightarrow \text{F}) \Rightarrow (\text{F})

    so we have a statement implying "false", so that statement is false.
    This bit here was massively helpful, cleared up my confusion regarding why I couldn't say x(x-1) \Rightarrow x=1. It makes a ton more sense when I add the "for all x s.t" before the sentence.

    Thank you!
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    (Original post by ghostwalker)
    Yes - added the AND and parentheses to avoid any confusion.
    Right, forgot about that - thanks!

    "x is an integer" is necessary by not sufficient.
    I see - so basically, I can kind of come to an intuitive grasp of this.

    x is an integer is necessary because whenever x(x-1) is true then "x is an integer is true" which is what makes it necessary but it isn't sufficient because you can have "x is an integer" but not "x(x-1) = 0" true is say x=5.

    Thanks! Can't rep you unfortunately.
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    (Original post by joostan)
    Yup that's it.



    Yes it would be sufficient but not necessary.

    Another good example of a sufficient but not necessary condition is that being divisible by 6 say, is sufficient but not necessary for a number being divisible by 3.
    On the other hand it is necessary, but not sufficient for a number to be divisible by 3 if it is divisible by 6.
    You often need to be careful with logic when proving things in a way that isn't emphasised at A-level or below, because obviously false statements can imply true statements.

    There are some other nice things you might like to look into, such as the contrapositive.
    Yeah, I'm always wary about proving things from false statements, ever since I read this:
    A story is told that the famous English mathematician G.H. Hardy made a remark at dinner that falsity implies anything. A guest asked him to prove that 2 + 2 = 5 implies that McTaggart is the Pope. Hardy replied, "We also know that 2 + 2 = 4,so that 5 = 4. Subtracting 3 we get 2 = 1. McTaggart and the Pope are two, hence McTaggart and the Pope are one."
    (not the true/full story, but it gets the point across and it was a quick google away)

    I'm trying to cement my understanding of basic things like necessity, and implications before moving on to things like contrapositive, which by the way, I can't wait to start investigating, there's a whole style of proof "proof by contraspositive" waiting to be discovered. :eek:

    On the other hand it is necessary, but not sufficient for a number to be divisible by 3 if it is divisible by 6.
    Right, so intuitively for me: that's a necessary condition because whenever a number is divisible by 6 then it is also divisible by 3 (but it's not a sufficient one because you can have a number that's divisible by 3 and not by 6)

    bracketed parts are where I'm dubious as to what I'm saying.

    Edit: Can't rep you either, PRSOM, thanks!
 
 
 
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