Exercises 1.1

1. List five elements in each of the following sets.

(a) {w  N : n is divisible by 5} (b) {2n+1:n  P}

(c) P({1,2.3.4,5}) (d) {2ⁿ:n  N}

(e) {l/n :n  P} (f) {r  Q:0 <r < 1}

(g) {n  N:n+1 is prime}

2. List the elements in the following sets.

(a) {l/n :n = l, 2, 3, 4} (b) {n²- n :n = 0, 1, 2, 3, 4]

(c) {l/n²:n  P, n is even and n < 11} (d) {2 + (-l)ⁿ :n  N}

3. List five elements in each of the following sets.

(a) ^* where  {a, b, c}

2. {w  ^length (w)  2} where  = {a, b}

3. {w  ^:length(w)=4} where {a, b}

Which sets above contain the empty word ?

4. Determine the following sets; i.e., list their elements if they are nonempty, and

write  if they are empty.

(a) {n  M:n² = 9} (b) {n  Z:n² = 9}

(c) {x  R:x² =9} (d) {n  N:3 < n < 7}

(e) {n  Z :3<|n|<7} (f) {x  R: x² < 0}

(g) {n  N:n² = 3} (h) {x  Q:x² = 3}

(i) {x  R :x < 1 and x  2} (j) {3n + 1 :n  N and n  6}

(k) {n  P :n is prime and n  15} [Recall that 1 isn't prime.]

5. How many elements are there in the following sets? Write  if the set is

infinite.

(a) {nN :n²=2} (b) {nZ:0  n 73}

(c) {nZ:5|n|73} (d) {nZ:5 < n < 73}

(e) {nZ :n is even and |n|  73} (f) {x e Q:0  x  73}

(g) {xQ : x² = 2} (h){xR :x² = 2}

(i) {x R:0.99 < x< 1.00) (j) P({0,1,2,3})

(k) P(N) (l) {n  N :n is even}

(m) {n N :n is prime} (n) {n  N :n is even and prime}

(o) {n  N: n is even or prime}

6. How many elements are there in the following sets? Write  if the set is

infinite.

(a) {-1, 1} (b)[-l, l]

(c)(-l, l) (d) {nZ:-l n  l}

(e) ^* where  = {a, b, c}

5. {w ^*:length(w) 4} where  = ({a, b, c}

7. Consider the sets

A = {n  P :n is odd}, B = (n  P :n is prime},

C = {4n + 3:nP}, D = {xR:x² - 8x + 15 = 0}.

Which of these sets are subsets of which? Consider all sixteen possibilities.

8. Consider the sets {0,1}, (0,1) and [0,1]. True or False.

1. {0, 1} (0, 1) (b) {0,1}  [0,1]

2. (0,1)  [0,1] (d) {0,l}  Z

(e) [0,1]  Z (f) [0,1]  Q

(g) 1/2 and /4 are in {0,1} (h) 1/2 and /4 are in (0,1)

(i) 1/2 and /4 are in [0,1]

9. Consider the following three alphabets: ₁ = {a, b, c}, ₂ = {a, b, ca} and ₃ = {a, b, Ab}. Determine to which of and - each word below belongs, and give its length as a member of each set to which it belongs.

(a) aba (b) bAb (c) cba

(d) cab (e) caab (f) baAb

10. Here is a question to think about. Let and imagine? if you can? a dictionary for all the nonempty words of with the words arranged in the usual alphabetical order. All the words a, aa, aaa, aaaa, etc. must appear before the word ba. How far into dictionary will you have to dig to find the word ba? How would the answer change if the dictionary contained only those words in of length 5 or less?

11. Suppose that w is a nonempty word in ^*.

(a) If the first [i.e., leftmost] letter of w is deleted, is the resulting string in ^*?

3. How about deleting letters from both ends of w? Are the resulting strings still

in ^*?

4. If you had a device that could recognize letters in  and could delete letters

from strings, how could you use it to determine if an arbitrary string of

symbols is in ^*?