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Instinctlol
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I am confused by what the book is saying, can someone explain how they got the thing I circled in red?
What don't you understand? The thing circled in red is explained just to the right of that.Instinctlol said:I am confused by what the book is saying, can someone explain how they got the thing I circled in red?
Instinctlol said:I am confused by what the book is saying, can someone explain how they got the thing I circled in red?
Proof by induction is a mathematical technique used to prove that a statement, known as a "predicate", is true for all natural numbers or integers. It involves establishing a base case, typically for n = 0 or n = 1, and then proving that if the statement holds for some arbitrary value of n, it also holds for n + 1. This process is repeated until it can be shown that the statement holds for all natural numbers or integers.
Proof by induction is unique in that it is specifically used to prove statements that are true for all natural numbers or integers. It relies on the principle of strong induction, which states that if a statement is true for all values less than or equal to some number, then it must also be true for the next number. This is different from other proof techniques, such as direct proof or proof by contradiction, which are used to prove specific cases or to disprove a statement.
Proof by induction is an important tool in mathematics and computer science, as it allows for the proof of statements that hold true for all natural numbers or integers. This is particularly useful in proving the correctness of algorithms or mathematical formulas. Understanding proof by induction also helps develop critical thinking skills and the ability to construct logical arguments.
One common mistake is assuming that the statement is true for a specific value of n without proving it for that value. This is known as "proof by example" and is not a valid form of proof by induction. Another mistake is starting the proof with n = 1 instead of n = 0, which can lead to incorrect conclusions. It is also important to make sure that the inductive step is valid and that the statement holds true for all values of n, not just a subset.
No, proof by induction can only be used to prove statements about natural numbers or integers. Real numbers are continuous and infinite, making it impossible to prove a statement for all real numbers using the principle of strong induction. Other proof techniques, such as direct proof or proof by contradiction, must be used for statements involving real numbers.