Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider the following three claims1. (n + k)...
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Consider the following three claims
1. (n + k)m = Θ(nm), where k and m are constants
2. 2n + 1 = O(2n)
3. 22n + 1 = O(2n)
2. 2n + 1 = O(2n)
3. 22n + 1 = O(2n)
Q. Which of these claims are correct ?
- a)1 and 2
- b)1 and 3
- c)2 and 3
- d)1, 2, and 3
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Consider the following three claims1. (n + k)m = Θ(nm), where k a...
(n + k)m and Θ(nm) are asymptotically same as theta notation can always be written by taking the leading order term in a polynomial expression. 2n + 1 and O(2n) are also asymptotically same as 2n + 1 can be written as 2 * 2n and constant multiplication/addition doesn't matter in theta notation. 22n + 1 and O(2n) are not same as constant is in power. See Asymptotic Notations for more details.
Most Upvoted Answer
Consider the following three claims1. (n + k)m = Θ(nm), where k a...
Claim 1: (n^k)^m = (n^m)
This claim is correct. According to the exponentiation rules, when you raise a power to another power, you can multiply the exponents. In this case, (n^k)^m can be written as n^(k*m), which is equal to n^m.
Claim 2: 2^n - 1 = O(2^n)
This claim is correct. In the big O notation, O(2^n) represents an upper bound on the growth rate of a function. The function 2^n - 1 grows exponentially, so it is definitely bounded by O(2^n) as the constant term (-1) does not affect the exponential growth.
Claim 3: 2^(2^n) - 1 = O(2^n)
This claim is incorrect. The function 2^(2^n) - 1 grows much faster than 2^n. When n increases, the value of 2^n grows exponentially, but the value of 2^(2^n) grows even faster. Therefore, 2^(2^n) - 1 cannot be bounded by O(2^n).
Conclusion:
Based on the explanations above, we can conclude that claim 1 and claim 2 are correct, while claim 3 is incorrect. Thus, the correct answer is option 'A' - 1 and 2.
This claim is correct. According to the exponentiation rules, when you raise a power to another power, you can multiply the exponents. In this case, (n^k)^m can be written as n^(k*m), which is equal to n^m.
Claim 2: 2^n - 1 = O(2^n)
This claim is correct. In the big O notation, O(2^n) represents an upper bound on the growth rate of a function. The function 2^n - 1 grows exponentially, so it is definitely bounded by O(2^n) as the constant term (-1) does not affect the exponential growth.
Claim 3: 2^(2^n) - 1 = O(2^n)
This claim is incorrect. The function 2^(2^n) - 1 grows much faster than 2^n. When n increases, the value of 2^n grows exponentially, but the value of 2^(2^n) grows even faster. Therefore, 2^(2^n) - 1 cannot be bounded by O(2^n).
Conclusion:
Based on the explanations above, we can conclude that claim 1 and claim 2 are correct, while claim 3 is incorrect. Thus, the correct answer is option 'A' - 1 and 2.
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Consider the following three claims1. (n + k)m = Θ(nm), where k and m are constants2. 2n + 1 = O(2n)3. 22n + 1 = O(2n)Q. Which of these claims are correct ?a)1 and 2b)1 and 3c)2 and 3d)1, 2, and 3Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following three claims1. (n + k)m = Θ(nm), where k and m are constants2. 2n + 1 = O(2n)3. 22n + 1 = O(2n)Q. Which of these claims are correct ?a)1 and 2b)1 and 3c)2 and 3d)1, 2, and 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following three claims1. (n + k)m = Θ(nm), where k and m are constants2. 2n + 1 = O(2n)3. 22n + 1 = O(2n)Q. Which of these claims are correct ?a)1 and 2b)1 and 3c)2 and 3d)1, 2, and 3Correct answer is option 'A'. Can you explain this answer?.
Consider the following three claims1. (n + k)m = Θ(nm), where k and m are constants2. 2n + 1 = O(2n)3. 22n + 1 = O(2n)Q. Which of these claims are correct ?a)1 and 2b)1 and 3c)2 and 3d)1, 2, and 3Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following three claims1. (n + k)m = Θ(nm), where k and m are constants2. 2n + 1 = O(2n)3. 22n + 1 = O(2n)Q. Which of these claims are correct ?a)1 and 2b)1 and 3c)2 and 3d)1, 2, and 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following three claims1. (n + k)m = Θ(nm), where k and m are constants2. 2n + 1 = O(2n)3. 22n + 1 = O(2n)Q. Which of these claims are correct ?a)1 and 2b)1 and 3c)2 and 3d)1, 2, and 3Correct answer is option 'A'. Can you explain this answer?.
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