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Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)?
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Suppose that P(n) is a propositional function. Determine for which pos...
**Solution:**
To determine for which positive integers n the statement P(n) must be true, we need to analyze the given conditions and apply them iteratively.
**Given conditions:**
1. P(1) is true.
2. For all positive integers n, if P(n) is true then P(n+2) is true.
**Analysis:**
Let's start by analyzing the first condition:
1. P(1) is true.
Since P(1) is true, it means that the statement P(n) is true for at least one positive integer n, specifically n = 1.
Next, let's analyze the second condition:
2. For all positive integers n, if P(n) is true then P(n+2) is true.
This condition implies that if P(n) is true for any positive integer n, then P(n+2) must also be true. In other words, if P(k) is true for some positive integer k, then P(k+2) must also be true.
Since we know that P(1) is true, we can apply the second condition to determine the truth of P(n) for subsequent positive integers.
**Iterative Application:**
Starting with n = 1, we can iteratively apply the second condition to determine the truth of P(n) for subsequent positive integers:
- P(1) is true (given).
- P(3) must be true (applying the second condition to n = 1).
- P(5) must be true (applying the second condition to n = 3).
- P(7) must be true (applying the second condition to n = 5).
- P(9) must be true (applying the second condition to n = 7).
- ...
**Conclusion:**
Based on the given conditions and the iterative application of the second condition, we can conclude that the statement P(n) must be true for all positive odd integers n.
Therefore, the correct answer is:
**Oa. P(3)**
To determine for which positive integers n the statement P(n) must be true, we need to analyze the given conditions and apply them iteratively.
**Given conditions:**
1. P(1) is true.
2. For all positive integers n, if P(n) is true then P(n+2) is true.
**Analysis:**
Let's start by analyzing the first condition:
1. P(1) is true.
Since P(1) is true, it means that the statement P(n) is true for at least one positive integer n, specifically n = 1.
Next, let's analyze the second condition:
2. For all positive integers n, if P(n) is true then P(n+2) is true.
This condition implies that if P(n) is true for any positive integer n, then P(n+2) must also be true. In other words, if P(k) is true for some positive integer k, then P(k+2) must also be true.
Since we know that P(1) is true, we can apply the second condition to determine the truth of P(n) for subsequent positive integers.
**Iterative Application:**
Starting with n = 1, we can iteratively apply the second condition to determine the truth of P(n) for subsequent positive integers:
- P(1) is true (given).
- P(3) must be true (applying the second condition to n = 1).
- P(5) must be true (applying the second condition to n = 3).
- P(7) must be true (applying the second condition to n = 5).
- P(9) must be true (applying the second condition to n = 7).
- ...
**Conclusion:**
Based on the given conditions and the iterative application of the second condition, we can conclude that the statement P(n) must be true for all positive odd integers n.
Therefore, the correct answer is:
**Oa. P(3)**
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Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)?
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Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)? for Computer Science Engineering (CSE) 2024 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 Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)?.
Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)? for Computer Science Engineering (CSE) 2024 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 Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) is true; for all positive integers n, if P(n) is true then P(n + 2) is true. Oa. P(3) Ob. P(4) OCP P(6) Od. P(2)?.
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