## Question 3

### (a) If A->B then A->>B

FALSE

Counterexample:

Grading:

3A -1.0 Invalid or missing counterexample.
3C -2.0 Rule incorrectly identified as T or F (regardless of the justification or counterexample)
Arguing with words did not count as a valid counterexample unless you
were sufficiently detailed and used variables to illustrate your arguments

### (b) If A->B and BC->D then AC->D

TRUE

You can justify this a variety of ways. Applying the closure to AC yields
ABCD (which contains D). The transitive rule also works. Note that you
**can't** reduce ABC->BD to AC->D by removing B from both sides - this
is invalid.

Grading:

3B -1.0 Invalid or missing justification.
3C -2.0 Rule incorrectly identified as T or F (regardless of the justification or counterexample)
### (c) If A->B and B->->C then A->->BD

TRUE

This one is a little more tricky. Applying the transitive rule to A->B
and B->C yields A->C. Applying the promotion rule to A->C then
yields A->->C. Applying the complementation rule to A->->C
then yields A->->BD. Some students presented less formal arguments
regarding what happens when (B,D) values are swapped between tuples
agreeing on A, given what you know about A->C. If clearly argued,
these solutions received credit as well.

Grading:

3B -1.0 Invalid or missing justification.
3C -2.0 Rule incorrectly identified as T or F (regardless of the justification or counterexample)