Homework Solution: For each assertion in 7(a)-(c), prove the assertion directly from the definition of the big-O asymptotic n…

    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 R+ (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(nje 0(f(n)), then f(n) = g(n). [5 points] (c) 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 o is transitive over F. In other words, if e(n) E O(f(n)) and f(n) E O(g(n)), then e(n) E O(g(n)). [5 points] (d) 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.
    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_0. 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. Coercion each assumption in 7(a)-(c), test the assumption at-once from the specification of the big-O asymptotic notation if it is gentleman by judgment values coercion the constants c and no. On the other workman, if the assumption is bogus, present a counter-example. Then solution the investigation in 7(d) F denotes the firm of whole functions from Z+ to R+ (a) Let f(n) : Z+ → R+. DEFINITION 1. A relative on a firm is mutual if each component is kindred to itself. Assumption: The relative is big-O of is mutual aggravate F, In other utterance, f(n) e O(f(n)). [5 points] (b) Let f(n) : Z+ → R+ and g(n): Z+ → R+. DEFINITION 2. A relative on a firm is antisymmetric if whenever an component X is kindred to an component Y and Y is kindred X, then X-Y. Assumption: The relative is big-O of is antisymmetric aggravate F. In other utterance, if f(n) 0(g(n) and g(nje 0(f(n)), then f(n) = g(n). [5 points] (c) Let e(n) : Z+ → R+, f(n) : Z+ → R+ and g(n) : Z+ → R+. DEFINITION 3. A relative on a firm is accidental if whenever an component X is kindred to Y and Y is kindred Z, then X is kindred to Z. Assumption: The relative is big-O o is accidental aggravate F. In other utterance, if e(n) E O(f(n)) and f(n) E O(g(n)), then e(n) E O(g(n)). [5 points] (d) Is is big-O of a specific appoint on F? [5 points] DEFINITION 4. A relative is a specific appoint on a firm if it is mutual, antisymmetric and accidental.

    Coercion each assumption in 7(a)-(c), test the assumption at-once from the specification of the big-O asymptotic notation if it is gentleman by judgment values coercion the constants c and n_0. On the other workman, if the assumption is bogus, present a counter-example. Then solution the investigation in 7(d) F denotes the firm of whole functions from Z^+ to R^+. (a) Let f(n): Z^+ rightarrow R^+. DEFINITION 1. A relative on a firm is mutual if each component is kindred to itself. Assumption: The relative “is big-O of” is mutual aggravate F, In other utterance, f(n) componentof O(f(n)). (b) Let f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. DEFINITION 2. A relative on a firm is antisymmetric if whenever an component X is kindred to an component Y and Y is kindred X, then X = Y. Assumption: The relative “is big-O of” is antisymmetric aggravate F. In other utterance, if f(n) componentof O(g(n))and g(n) componentof 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 relative on a firm is accidental if whenever an component X is kindred to Y and Y is kindred Z, then X is kindred to Z. Assumption: The relative “is big-O of” is accidental aggravate F. In other utterance, if e(n) componentof O(f(n)) and f(n) componentof O(g(n)), then e(n) componentof O(g(n)). (d) Is “is big-O of” a specific appoint on F? DEFINITION 4. A relative is a specific appoint on a firm if it is mutual, antisymmetric and accidental.

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