Theorem (The Principle of Mathematical Induction) For each positive integer n, let P(n) be a statement. If (1) P(1) is true and (2) the implication If P(k), then P(k + 1). is true for every positive integer k, then P(n) is true for every positive integer n. Proof Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied but there exist some positive integers n for which P(n) is a false statement. Let S = {n EN: P(n) is false}. Since S is a nonempty subset of N, it follows by the Well-Ordering Principle that S con- tains a least element s. Since P(1) is true, 1 # S. Thus, s≥ 2 and s − 1 € N. Therefore, s – 1 Sand so P(s − 1) is a true statement. By condition (2), P(s) is also true and so s & S. This, however, contradicts our assumption that s € S.
Theorem (The Principle of Mathematical Induction) For each positive integer n, let P(n) be a statement. If (1) P(1) is true and (2) the implication If P(k), then P(k + 1). is true for every positive integer k, then P(n) is true for every positive integer n. Proof Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied but there exist some positive integers n for which P(n) is a false statement. Let S = {n EN: P(n) is false}. Since S is a nonempty subset of N, it follows by the Well-Ordering Principle that S con- tains a least element s. Since P(1) is true, 1 # S. Thus, s≥ 2 and s − 1 € N. Therefore, s – 1 Sand so P(s − 1) is a true statement. By condition (2), P(s) is also true and so s & S. This, however, contradicts our assumption that s € S.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please explain the proof in more detail step by step, I really dont understand a word of it.
Transcribed Image Text:Theorem
(The Principle of Mathematical Induction) For each positive integer n, let P(n) be a
statement. If
(1) P(1) is true and
(2)
the implication
If P(k), then P(k + 1).
is true for every positive integer k,
then P(n) is true for every positive integer n.
Proof Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied
but there exist some positive integers n for which P(n) is a false statement. Let
S = {n EN: P(n) is false}.
Since S is a nonempty subset of N, it follows by the Well-Ordering Principle that S con-
tains a least element s. Since P(1) is true, 1 # S. Thus, s ≥ 2 and s - 1 € N. Therefore,
s - 1 S and so P(s − 1) is a true statement. By condition (2), P(s) is also true and so
s & S. This, however, contradicts our assumption that s € S.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
