answer the following questions. n-5 n is 1. We have a function f: Z Z...

  

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answer the following questions. n-5 n is 1. We have a function f: Z → Z defined as f(n) = {n+3 if n is odd. Find whether the function is injective and surjective. What would be the inverse of this function? 2. Consider three sets A, B, and C each with three elements, and define a function 'f from set A to B and a function 'g' from set B to C with the mapping of the elements you specify. After defining 'f' and 'g', discuss the composition of functions fog and gof on the sets A, B, and C that you have defined. Can both fog and gof be defined for the sets you have considered? Are they equal? Provide proof to support your answer. 3. There are 5 students; Ani, Leon, Linh, Liam, and Abdul with scores in computer science as 75, 60, 85, 95, and 60 respectively. They are graded by the University as B, C, B+, A, and C respectively. Define the mappings from students to marks and marks to grades explicitly. What are the domains and ranges of the functions? Can you make a composite function out of this? If yes, make the composite function and find if the composition of functions is commutative. Explain the reason. 4. Explain the following concepts using relevant examples: sequence, recursive function, closed formula for the sequence, arithmetic sequence, and geometric sequence. Ensure that you do not take the same examples discussed in the textbooks and online resources. 5. i. The sequence 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12........ can be generated by a closed formula. Find the formula and the next four terms of the sequence. ii. Find the next three terms of the series, provide a closed formula and calculate the sum to the first n terms for the following series: 6+36+216+................. iii. Find the next three terms of the series, provide a closed formula, and calculate the sum to the first n terms for the following series: 21+24+27+.. answer the following questions. n-5 n is 1. We have a function f: Z → Z defined as f(n) = {n+3 if n is odd. Find whether the function is injective and surjective. What would be the inverse of this function? 2. Consider three sets A, B, and C each with three elements, and define a function 'f from set A to B and a function 'g' from set B to C with the mapping of the elements you specify. After defining 'f' and 'g', discuss the composition of functions fog and gof on the sets A, B, and C that you have defined. Can both fog and gof be defined for the sets you have considered? Are they equal? Provide proof to support your answer. 3. There are 5 students; Ani, Leon, Linh, Liam, and Abdul with scores in computer science as 75, 60, 85, 95, and 60 respectively. They are graded by the University as B, C, B+, A, and C respectively. Define the mappings from students to marks and marks to grades explicitly. What are the domains and ranges of the functions? Can you make a composite function out of this? If yes, make the composite function and find if the composition of functions is commutative. Explain the reason. 4. Explain the following concepts using relevant examples: sequence, recursive function, closed formula for the sequence, arithmetic sequence, and geometric sequence. Ensure that you do not take the same examples discussed in the textbooks and online resources. 5. i. The sequence 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12........ can be generated by a closed formula. Find the formula and the next four terms of the sequence. ii. Find the next three terms of the series, provide a closed formula and calculate the sum to the first n terms for the following series: 6+36+216+................. iii. Find the next three terms of the series, provide a closed formula, and calculate the sum to the first n terms for the following series: 21+24+27+..

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1 Function f Z Z fn n 3 if n is odd fn n 5 if n is even Injective A function is injective or onetoone if each element of the domain maps to a distinct element in the codomain In this case the function ... View the full answer