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IDC
Computability and Complexity Theory
R.Zviel-Girshin
L.Epstein
Exercise 1 Solution
Question 1(12)
Write a short English/Hebrew description of each set.
{-3,-6,-9,-12,...}= Set of negative natural numbers that divides by 3 with no
remainder excluding 0.
{1,2,3,4,..}= Set of all natural numbers excluding 0.
{n| n=3m for some mN}= Set of natural numbers that divides by 3 with no
remainder.
{n|n=2i+3j for some iZ and jZ}= Set of linear combination of 2 and 3
with positive coefficients.
{n|n=2i+3j for some iN and jN}= Set of all linear combination of 2 and 3.
Question 2(15)
Write formal descriptions of the following sets.
The set of all negative odd numbers
= { 2n+1|nZ- }= { -2n+1|nZ }
The set containing all palindromes of length 3 over ={0,1,2}
= {w| w=xyx, x,y  {0,1,2} }
The set containing all prefixes and suffixes of the word abbab
= {ε, a, ab, abb, abba, abbab, b, bab, bbab }
Question 3(25)
Determine whether each of the following is true or false (a short explanation
is required).
1.
2.
3.
4.
5.
6.

False, the empty set has no elements.

True, every set is inclusive to itself.
{}
True, the element from the left exists in the right set.
{}
True, the empty set is inclusive to any set.
{a,b,{a,b}} - {a,b}={a,b} False
{a,b}2{a,b,{b,a}}
True, {a,b} exists in the power set of {a,b,{b,a}}
‫גירשין‬-‫רינה צביאל‬
IDC
Computability and Complexity Theory
R.Zviel-Girshin
L.Epstein
Question 4(24)
Given a word w=abcbbbca.
Determine whether each of the following is true or false.
1. bcb is a substring of w.
True
2. bcc is a substring of w.
False
3. abc is a prefix of w. True
4. bbca is a suffix of w.
True
5. The number of substrings of abcb equals to the number of substrings
of bbca. True
Answer the following questions:
6. If u=abc and w=uv, then what is v? bbbca
7. How many prefixes of w are also suffixes of w? 3 (ε, a, w)
8. What is wr? acbbbcba
Question 5(24)
Given the following languages over ={a,b}
L1= {}
L2 = 
L3 = { abc, a, bb }
L4 = {aa, bb,  }
L5 = { a, bb, aaa, aba }
Compute the resulting language of the following operations:
1. L1R = {ε}
2. L12 L5= {a, bb, aaa, aba}
3. L32= {abcabc, abca, abcbb, aabc, aa, abb, bbabc, bba, bbbb}
4. L5R= {a, bb, aaa, aba} = L5
5. L3 L5R= {abca, abcbb, abcaaa, abcaba, aa, abb, aaaa, aaba, bba,
bbbb, bbaaa, bbaba}
6. L4 L3= {aaabc, aaa, aabb, bbabc, bba, bbbb, abc, a, bb}
7. L42= {aaaa, aabb, aa, bbaa, bbbb, bb}
8. L3  L2= 
9. L5 - L4R = L5 – L4 = {a, aaa, aba}
10. L3  L4= {abc, a, bb, aa, ε}
11. L5 L2  L30= 
12. (L3  L5)R= {abc, a, bb, aaa, aba}R = {cba, a, bb, aaa, aba}
13. L1*= {ε}
14. L1+= {ε}
15. L2*= 
16. L2+= 
17. L4*= {L04  L14  L24  }
‫גירשין‬-‫רינה צביאל‬