## Question 3

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

FALSE

Counterexample:
A B C D
0 1 3 4
0 2 3 4

• 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.

• 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.