Re: ?? Functional Dependency Question ??
From: paul c <toledobythesea_at_oohay.ac>
Date: Mon, 20 Oct 2008 00:35:56 GMT
Message-ID: <MnQKk.2548$fF3.942_at_edtnps83>
>> tlbaxte..._at_yahoo.com wrote:
>>
>>> "Although X->A and X->B implies X->AB by the union rule stated above,
>>> X->A, and Y->B does *not* imply that XY->AB."
>>> I'm not seeing this. It seems to me that X->A, and Y->B *DOES* imply
>>> that XY->AB.
>>> I'm sure I'm wrong but I'm not seeing it. Can someone explain?
>>> Thanks
>> I'm not seeing it either. By these truth tables, it seems to:
>>
>> XY AB X->A Y->B (X->A)(Y->B) XY->AB (X->A)(Y->B)->(XY->AB)
>> 00 00 1 1 1 1 1
>> 00 01 1 1 1 1 1
>> 00 11 1 1 1 1 1
>> 00 10 1 1 1 1 1
>> 01 10 1 0 0 1 1
>> 01 11 1 1 1 1 1
>> 01 01 1 1 1 1 1
>> 01 00 1 0 0 1 1
>> 11 00 0 0 0 0 1
>> 11 01 0 1 0 0 1
>> 11 11 1 1 1 1 1
>> 11 10 1 0 0 0 1
>> 10 10 1 1 1 1 1
>> 10 11 1 1 1 1 1
>> 10 01 0 1 0 1 1
>> 10 00 0 1 0 1 1
>>
>> (View with a fixed width font)
>>
>> Can anyone find a mistake in the above truth tables? Is there a
>> difference between functional dependency and implication that I need to
>> learn?
Date: Mon, 20 Oct 2008 00:35:56 GMT
Message-ID: <MnQKk.2548$fF3.942_at_edtnps83>
Keith H Duggar wrote:
> On Oct 18, 11:18 am, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
>> tlbaxte..._at_yahoo.com wrote:
>>
>>> "Although X->A and X->B implies X->AB by the union rule stated above,
>>> X->A, and Y->B does *not* imply that XY->AB."
>>> I'm not seeing this. It seems to me that X->A, and Y->B *DOES* imply
>>> that XY->AB.
>>> I'm sure I'm wrong but I'm not seeing it. Can someone explain?
>>> Thanks
>> I'm not seeing it either. By these truth tables, it seems to:
>>
>> XY AB X->A Y->B (X->A)(Y->B) XY->AB (X->A)(Y->B)->(XY->AB)
>> 00 00 1 1 1 1 1
>> 00 01 1 1 1 1 1
>> 00 11 1 1 1 1 1
>> 00 10 1 1 1 1 1
>> 01 10 1 0 0 1 1
>> 01 11 1 1 1 1 1
>> 01 01 1 1 1 1 1
>> 01 00 1 0 0 1 1
>> 11 00 0 0 0 0 1
>> 11 01 0 1 0 0 1
>> 11 11 1 1 1 1 1
>> 11 10 1 0 0 0 1
>> 10 10 1 1 1 1 1
>> 10 11 1 1 1 1 1
>> 10 01 0 1 0 1 1
>> 10 00 0 1 0 1 1
>>
>> (View with a fixed width font)
>>
>> Can anyone find a mistake in the above truth tables? Is there a
>> difference between functional dependency and implication that I need to
>> learn?
> > Your truth table is correct. You can also prove this with > Boolean algebra (below ~ = not, + = or, * = and): > > given : > > (1) 1 = ~X + A : X implies A > (2) 1 = ~Y + B : Y implies B > > prove : > > (3) 1 = ~(XY) + AB : XY implies AB > > proof : > > (4) 1 = (~X + A)(~Y + B) : conjuction of (1) and (2) > (5) 1 = ~X~Y + ~XB + ~YA + AB : distributive and commutative > (6) 1 = ~X~Y + ~X~Y + ~XB + ~YA + AB : idempotent > (7) 1 = ~X(~Y + B) + ~Y(~X + A) + AB : distributive and commutative > (8) 1 = ~X + ~Y + AB : substitute (1) and (2) > (9) 1 = ~(XY) + AB : De Morgan > > QED > > KHD >
Nice, step 6 seems cute (to an amateur like me). It's seeing the parallels that fascinates me, seeing parallels seems to be important in the RM, perhaps more important than seeing ways to express structure in precise abstract terms, eg mathematical terms, as opposed to graphical or visual ways or ways that resort to particular imprecise programming methods. Received on Mon Oct 20 2008 - 02:35:56 CEST