Semester Problem Set 2 Discrete Math Solutions
ATILIM UNIVERSITY DEPARTMENT OF MATHEMATICS Math 211 - Discrete Mathematics with Applications 2010-2011 Fall Semester Problem Set I Prepared by Mehmet TURAN
O−, Ω−, Θ− Notations
1. Let f and g be real valued functions defined on the same set of nonnegative real numbers. (a) Prove that if g(x) is O(f (x)), then f (x) is Ω(g(x)). (b) Prove that if f (x) is O(g(x)) and c is any nonzero ral number, then cf (x) is O(cg(x)). (c) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x) + g(x) is O(G(x)), where, for each x in the domain, G(x) = max(|h(x)|, |k(x)|). (d) Prove that f (x) is Θ(f (x)). (e) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x)g(x) is O(h(x)k(x)). 2. (a) Show that for any real number x, if x > 1 then |x3
| ≤ |2x3 + 3x + 4|. (b) Show that for any real number x, if x > 1 then |2x3 + 3x + 4| ≤ 9|x3 |. (c) Use the Ω− and O− notations to express the results of parts (a) and (b). (d) What can you deduce about the order of 2x3 + 3x + 4? 3. Use the definition of Θ− notation to show that 3x5 + 7x2 + 5 is Θ(x5 ).
1 4. (a) Show that for any real number x, if x > 1 then | 2 x2 − 14x + 5| ≤ 20|x2 |.
(b) Use the O− notation to express the result of part (a). 5. (a) Show that for any real number x, if x > 1 then | 1 x5 − 3x3 + 2x − 5| ≤ 11|x5 |. 3 (b) Use the O− notation to express the result of part (a). 6. Show that x4 is not O(x2 ). 7. Use the definition of Ω− notation to show that 6x3 − 13x − 5 is Ω(x3 ). 8. Use the definition of Ω− notation to show that 9. Let n be a positive integer. (a) Show that 12 + 22 + 32 + · · · + n2 is Θ(n3 ). (b) Show that 13 + 23 + 33 + · · · + n3 is Θ(n4 ). 1
1 2 2x
− 3x − 5 is Ω(x2 ).
(c) Show that 2 + 4 + 6 + · · · + 2n is Θ(n2 ). n (d) Show that i=1 n
(6i − 5) is Θ(n2 ). i(i + 3) is Θ(n3 ). i=1 n
(e) Show that
(f) Show that k=1 (k 2 − 2k) is Θ(n3 ).
10. Use the inequality n < 2n to show that log n is O(n).
2
O−, Ω−, Θ− Notations
1. Let f and g be real valued functions defined on the same set of nonnegative real numbers. (a) Prove that if g(x) is O(f (x)), then f (x) is Ω(g(x)). (b) Prove that if f (x) is O(g(x)) and c is any nonzero ral number, then cf (x) is O(cg(x)). (c) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x) + g(x) is O(G(x)), where, for each x in the domain, G(x) = max(|h(x)|, |k(x)|). (d) Prove that f (x) is Θ(f (x)). (e) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x)g(x) is O(h(x)k(x)). 2. (a) Show that for any real number x, if x > 1 then |x3
| ≤ |2x3 + 3x + 4|. (b) Show that for any real number x, if x > 1 then |2x3 + 3x + 4| ≤ 9|x3 |. (c) Use the Ω− and O− notations to express the results of parts (a) and (b). (d) What can you deduce about the order of 2x3 + 3x + 4? 3. Use the definition of Θ− notation to show that 3x5 + 7x2 + 5 is Θ(x5 ).
1 4. (a) Show that for any real number x, if x > 1 then | 2 x2 − 14x + 5| ≤ 20|x2 |.
(b) Use the O− notation to express the result of part (a). 5. (a) Show that for any real number x, if x > 1 then | 1 x5 − 3x3 + 2x − 5| ≤ 11|x5 |. 3 (b) Use the O− notation to express the result of part (a). 6. Show that x4 is not O(x2 ). 7. Use the definition of Ω− notation to show that 6x3 − 13x − 5 is Ω(x3 ). 8. Use the definition of Ω− notation to show that 9. Let n be a positive integer. (a) Show that 12 + 22 + 32 + · · · + n2 is Θ(n3 ). (b) Show that 13 + 23 + 33 + · · · + n3 is Θ(n4 ). 1
1 2 2x
− 3x − 5 is Ω(x2 ).
(c) Show that 2 + 4 + 6 + · · · + 2n is Θ(n2 ). n (d) Show that i=1 n
(6i − 5) is Θ(n2 ). i(i + 3) is Θ(n3 ). i=1 n
(e) Show that
(f) Show that k=1 (k 2 − 2k) is Θ(n3 ).
10. Use the inequality n < 2n to show that log n is O(n).
2