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The principle of mathematical induction is for the set of:
- a)Rational numbers
- b)Positive integers
- c)Whole number
- d)Integers
Correct answer is 'B'. Can you explain this answer?
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The principle of mathematical induction is for the set of:a)Rational n...
A class of integers is called hereditary if, whenever any integer x belongs to the class, the successor of x (that is, the integer x + 1) also belongs to the class. The principle of mathematical induction is then: If the integer 0 belongs to the class F and F is hereditary, every nonnegative integer belongs to F.
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The principle of mathematical induction is for the set of:a)Rational n...
Principle of Mathematical Induction
Mathematical induction is a powerful method used to prove statements about positive integers. The principle of mathematical induction is specifically designed for the set of positive integers.
Explanation
Base Case:
- The first step in using mathematical induction is to prove that the statement holds true for the base case, which is usually the first positive integer (often 1 or 0).
Inductive Hypothesis:
- Assume that the statement is true for some positive integer k. This is known as the inductive hypothesis.
Inductive Step:
- The next step is to prove that if the statement is true for k, then it must also be true for k+1. This is known as the inductive step.
Conclusion:
- By proving the base case and showing that the statement holds for k implies it holds for k+1, we can conclude that the statement is true for all positive integers.
Why it applies to positive integers:
- The principle of mathematical induction specifically applies to positive integers because it relies on the concept of a successor. It starts with a base case and then builds upon that by showing that if the statement is true for a particular positive integer k, it must also be true for the next positive integer k+1. This process is unique to the set of positive integers and is not directly applicable to other sets such as rational numbers or integers.
Mathematical induction is a powerful method used to prove statements about positive integers. The principle of mathematical induction is specifically designed for the set of positive integers.
Explanation
Base Case:
- The first step in using mathematical induction is to prove that the statement holds true for the base case, which is usually the first positive integer (often 1 or 0).
Inductive Hypothesis:
- Assume that the statement is true for some positive integer k. This is known as the inductive hypothesis.
Inductive Step:
- The next step is to prove that if the statement is true for k, then it must also be true for k+1. This is known as the inductive step.
Conclusion:
- By proving the base case and showing that the statement holds for k implies it holds for k+1, we can conclude that the statement is true for all positive integers.
Why it applies to positive integers:
- The principle of mathematical induction specifically applies to positive integers because it relies on the concept of a successor. It starts with a base case and then builds upon that by showing that if the statement is true for a particular positive integer k, it must also be true for the next positive integer k+1. This process is unique to the set of positive integers and is not directly applicable to other sets such as rational numbers or integers.
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