Homework Solution: For each assertion in 7(a)-(c), prove the assertion directly from the definition of the big-O asymptotic notation if it is true by finding val…

    7. For each assertion in 7(a)-(c), prove the assertion directly from the definition of the big-O asymptotic notation if it is true by finding values for the constants c and no. On the other hand, if the assertion is false, give a counter-example. Then answer the question in 7(d). F denotes the set of all functions from Z+ to Rt (a) Let f(n) : Z+ → R+. DEFINITION 1. A relation on a set is reflexive if each element is related to itself. Assertion: The relation is big-O of is reflexive over F, In other words, f(n) E O(f(n)). [5 points] (b) Let f(n) : Z+ → R+ and g(n) : Z+ → R+ DEFINITION 2. A relation on a set is antisymmetric if whenever an element X is related to an element Y and Y is related X, then X Y. Assertion: The relation is big-O of is antisymmetric over F. In other words, if f(n) 0(g(n)) and g(n) ε o(f(n)), then f(n)-g(n). [5 points] Let e(n) : Z+ → R+, f(n) : Z+ → R+ and g(n) : Z+ → R+. DEFINITION 3. A relation on a set is transitive if whenever an element X is related to Y and Y is related Z, then X is related to Z. Assertion: The relation is big-O of is transitive over F In other words, if e(n) E O((n)) and f(n) E O(g(n)), then e(n) e O(g(n)). [5 points] Is is big-O of a partial order on F? [5 points] DEFINITION 4. A relation is a partial order on a set if it is reflexive, antisymmetric and transitive. (c) (d)
    For each assertion in 7(a)-(c), prove the assertion directly from the definition of the big-O asymptotic notation if it is true by finding values for the constants c and n_o. On the other hand, if the assertion is false, give a counter-example. Then answer the question in 7(d). F denotes the set of all functions from Z^+ to R^+. (a) Let f(n): Z^+ rightarrow R^+. Definition 1. A relation on a set is reflexive if each element is related to itself. Assertion: The relation "is big-O of" is reflexive over F, In other words, f(n) elementof O (f(n)). (b) Let f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. Definition 2. A relation on a set is antisymmetric if whenever an element X is related to an element Y and Y is related X, then X = Y. Assertion: The relation "is big-O of" is antisymmetric over F. In other words, if f(n) elementof O(g(n)) and g(n) elementof O(f(n)), then f(n) = g(n). (c) Let e(n): Z^+ rightarrow R^+, f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. Definition 3. A relation on a set is transitive if whenever an element X is related to Y and Y is related Z, then X is related to Z. Assertion: The relation "is big-O of" is transitive over F. In other words, if e(n) elementof O(f(n)) and f(n) elementof O(g(n)), then e(n) elementof O(g(n)). (d) Is "is big-O of" a partial order on F? Definition 4. A relation is a partial order on a set if it is reflexive, antisymmetric and transitive.

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    7. Ce each assumption in 7(a)-(c), ascertain the assumption quickly from the restriction of the big-O asymptotic notation if it is gentleman by sentence values ce the constants c and no. On the other operative, if the assumption is faithless, produce a counter-example. Then acceptance the inquiry in 7(d). F denotes the coercionmal of entire functions from Z+ to Rt (a) Let f(n) : Z+ → R+. DEFINITION 1. A kinsman on a coercionmal is alternate if each part is cognate to itself. Assumption: The kinsman is big-O of is alternate balance F, In other articulation, f(n) E O(f(n)). [5 points] (b) Let f(n) : Z+ → R+ and g(n) : Z+ → R+ DEFINITION 2. A kinsman on a coercionmal is antisymmetric if whenever an part X is cognate to an part Y and Y is cognate X, then X Y. Assumption: The kinsman is big-O of is antisymmetric balance F. In other articulation, if f(n) 0(g(n)) and g(n) ε o(f(n)), then f(n)-g(n). [5 points] Let e(n) : Z+ → R+, f(n) : Z+ → R+ and g(n) : Z+ → R+. DEFINITION 3. A kinsman on a coercionmal is explicit if whenever an part X is cognate to Y and Y is cognate Z, then X is cognate to Z. Assumption: The kinsman is big-O of is explicit balance F In other articulation, if e(n) E O((n)) and f(n) E O(g(n)), then e(n) e O(g(n)). [5 points] Is is big-O of a biased manage on F? [5 points] DEFINITION 4. A kinsman is a biased manage on a coercionmal if it is alternate, antisymmetric and explicit. (c) (d)

    Ce each assumption in 7(a)-(c), ascertain the assumption quickly from the restriction of the big-O asymptotic notation if it is gentleman by sentence values ce the constants c and n_o. On the other operative, if the assumption is faithless, produce a counter-example. Then acceptance the inquiry in 7(d). F denotes the coercionmal of entire functions from Z^+ to R^+. (a) Let f(n): Z^+ rightarrow R^+. Restriction 1. A kinsman on a coercionmal is alternate if each part is cognate to itself. Assumption: The kinsman “is big-O of” is alternate balance F, In other articulation, f(n) partof O (f(n)). (b) Let f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. Restriction 2. A kinsman on a coercionmal is antisymmetric if whenever an part X is cognate to an part Y and Y is cognate X, then X = Y. Assumption: The kinsman “is big-O of” is antisymmetric balance F. In other articulation, if f(n) partof O(g(n)) and g(n) partof O(f(n)), then f(n) = g(n). (c) Let e(n): Z^+ rightarrow R^+, f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. Restriction 3. A kinsman on a coercionmal is explicit if whenever an part X is cognate to Y and Y is cognate Z, then X is cognate to Z. Assumption: The kinsman “is big-O of” is explicit balance F. In other articulation, if e(n) partof O(f(n)) and f(n) partof O(g(n)), then e(n) partof O(g(n)). (d) Is “is big-O of” a biased manage on F? Restriction 4. A kinsman is a biased manage on a coercionmal if it is alternate, antisymmetric and explicit.

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