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CS154 Spring 2006Practice Midterm 1

To study for the midterm I would suggest you: (1) Know how to do (by heart) all the practice problems. (2) Go over your notes at least three times. Second and third time try to see how much you can remember from the first time. (3) Go over the homework problems. (4) Try to create your own problems similar to the ones I have given and solve them. (5) Skim the relevant sections from the book. (6) If you want to study in groups, at this point you are ready to quiz each other. The practice midterm is below. Here are some facts about the actual midterm: (a) The midterm will be in class Feb 20. (b) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (c) You should bring photo ID. (d) There will be more than one version of the test. Each version will be of comparable difficulty. (e) If your cell-phone or beeper goes off you will be excused from the test at that point and graded on what you have done till your excusal. (f) One problem (less typos) on the actual test will be from the practice test.

[Practice Midterm1 Student Solutions-PDF]

1. Prove there is a bijection between the natural numbers and the even numbers

2. What is the power set of {1, 2, 3}?

3. Call two languages L1 and L2 similar if they differ by only finite sets. Prove this notion of similar is an equivalence relation.

4. Give an example of a tree, a graph with and without a cycle, and a graph which is connected as well as one that is not.

5. Prove every 2-ary Boolean function can constructed from (A NOR B) := NOT(A OR B) using just composition.

6. Prove by induction that (n)!/(n-k)!k! = (n-1)!/(n-k)!k-1! + (n-1)!/(n-1-k)!k!

7. Use the cartesian product construction to construct an automata that recognizes strings over {0,1} whose length is either a multiple of 4 or a multiple of 5 from automata for each of these languages separately.

8. Let A+ = ∪k>0Ak. Prove the regular languages are closed under this operation.

9. Use the construction from the book to create an NFA recognizing bb(a∪b)*a.

10. Create an NFA for the language {0w1 | w is a string over {0,1}}.Then show how the construction from the book would convert this NFA to a regular expression