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So let's see if it does; i.e., let's apply the given dependencies.
From A C , we have y13 = y23 = y33 ; likewise, from B C , we have y13 = y43 . So we can replace each
of y23 , y33 , and y43 by y13 . The premise tuples become (replacements shown in bold ):
x1 x2 y13 y14 y15
x1 y22 y13 x4 y25
x1 y32 y13 y34 x5
y41 x2 y13 y44 x5
y51 y52 x3 x4 x5
From C D , we have y14 = y34 = y44 = x4 . Make the replacements:
x1 x2 y13 x4 y15
x1 y22 y13 x4 y25
x1 y32 y13 x4 x5
y41 x2 y13 x4 x5
y51 y52 x3 x4 x5
From CE A , we have y41 = x1 . Make the replacements:
x1 x2 y13 x4 y15
x1 y22 y13 x4 y25
x1 y32 y13 x4 x5
x1 x2 y13 x4 x5
y51 y52 x3 x4 x5
From DE C , we have y13 = x3 . Make the replacements:
x1 x2 x3 x4 y15
x1 y22 x3 x4 y25
x1 y32 x3 x4 x5
x1 x2 x3 x4 x5 ◄═ Success: all x's !!!
y51 y52 x3 x4 x5
The “fourth” tuple here is all x 's, and so the JD J does indeed follow from the given FDs.
Let's look at another example. Let the given set of dependencies consist of just the JD { AB , AC }. Does this
set imply the FD A B ? Note: We already know the answer is no, because what we're talking about here is the
converse of Heath's Theorem, and we know from Exercise 5.4 that the converse of Heath's Theorem is false. But
let's see what the chase tells us:
Premise tuples:
x1 y12 y13
x1 y22 y23
If and only if the FD is implied by the JD, then applying the JD to these tuples will have to make y12 and y22
equal. Does it do so? Well:
The given JD “generates” tuples as follows:

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