Download Exercises Warm Up to the Theory of Computation

Exercises Chapter 0, Sipser Book
Warm Up to the Theory of Computation
1. For each of the first 3 lines in the table below, construct a relation over the set {1, 2, 3} that
fits the description.
2. Describe each of the following infinite sets precisely, using a formula that does not involve
“…”. If you wish, you can use N, R,Q, Z, and common notations for sets.
a.
{0, -1, 2, -3, 4, -5, . . .}
b.
{1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, 3/16, 5/16, 7/16, . . .}
c.
{10, 1100, 111000, 11110000, . . .} ( a subset of {0,1}* )
d.
{{0}, {1}, {2}, . . .}
e.
{{0}, {0,1}, {0, 1, 2}, {0, 1, 2, 3}, . . .}
3. Let L1 and L2 be subsets of {a, b}*, and consider the two languages
L1* È L2* and (L1 È L2)*.
Which of the two is always a subset of the other? Give an example (i.e., say what L1 and L2
are) so that the opposite inclusion does not hold.
4. Sipser, Problem 0.1
5. Sipser, Problem 0.2
6. Sipser, Problem 0.8
7. Sipser, Problem 0.9
------------------------------------8. Prove that the square root of 8 is irrational.
9. Prove or disprove the following statements:
(a) The sum of a rational and an irrational number must be irrational.
(b) The sum of two positive irrational numbers must be irrational.
(c) The product of two irrational numbers must be irrational.
10. Use contradiction to show there are no positive integer solutions to the equation
2
2
x - y = 1.
11. Prove by construction that there are infinitely many solutions in positive integers x, y, and z
to the equation
2
2
x + y =z
2
12. Prove using induction
∑i2 = k(k + 1) (2k + 1) / 6
13. Prove using induction
n! > 2n n >= 4
(from i = 1 to k)