The process of reasoning that a rule, condition, definition, property, or statement is true because specific cases have been observed to be true is called ___.
The process of reasoning that you are referring to is called induction
The process of reasoning that you are referring to is called induction. Induction is a method of logical reasoning where you observe specific cases or examples and use them to make a general conclusion or prediction. It involves making a hypothesis or generalization based on the observed patterns or behaviors of specific instances.
In the context of mathematics, the process of induction is commonly used to prove statements or propositions known as mathematical induction. Mathematical induction is a way to establish the truth of a statement that holds for infinitely many cases, usually involving a sequence or series of numbers.
The process of mathematical induction generally follows three steps:
1. Base Case: The first step is to prove that the statement holds true for a specific value, usually the smallest value or the starting point of a sequence. This is known as the base case.
2. Inductive Hypothesis: The second step is to assume that the statement holds true for a certain value, usually denoted as ‘k’, representing an arbitrary case.
3. Inductive Step: The final step is to prove that if the statement holds true for the ‘k’ case, it also holds true for the next case, which is ‘k+1’. This is known as the inductive step.
By proving that the statement holds true for both the base case and proving that if it holds for one case, it holds for the next, we can conclude that the statement holds true for all values beyond the base case.
Induction is a powerful tool in mathematics as it allows us to establish general truths based on specific observations. It is commonly used in various branches of mathematics, including number theory, algebra, and calculus, to prove theorems and establish important mathematical properties.
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