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Expert-verified Found in: Page 329 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # There are infinitely many stations on a train route. Sup-pose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that the train stops at all stations.

Examples of Hypothesis In the conditional, "If all four sides of a quadrilateral measure the same, then the quadrilateral is a square" the hypothesis is "all four sides of a quadrilateral measure the same".

See the step by step solution

## Step 2

Let p(k) be the hypothesis that the train stops at station k.

Base case: P(1) is true because we are told that the train stops at the first station.

Induction step: Assume that the train stops at station k - 1 that is P ( k - 1 ) , is true.

Now, we are told to assume that, if the train stops at a station, then it stops at the next station. Since we assumed that it stopped at station k - 1 (by induction hypothesis), it means that it will stop at station k. Thus P (K) , is true.

Then, by induction, P(k) is true for all k and, therefore, the train stops at all stations. ### Want to see more solutions like these? 