Problem 1E

Use mathematical induction (and the proof of Proposition as a model) to show that any amount of money of at least 14c can be made up using 3¢ and 8¢ coins.

Proposition

For all integers n ≥ 8, n¢ can be obtained using 3¢ and ¢ coins.

Proof (by mathematical induction):

Let the property P(n) be the sentence

n¢ can be obtained using 3¢ and 5¢coins. ← P(n)

Show that P(8) is true:

P(8) is true because 8¢can be obtained using one 3¢coin and one 5¢ coin.

Show that for all integers k≥ 8, if P(k) is true then P(k+1) is also true:

[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ≥ 8. That is:]

Suppose that k is any integer with k ≥ 8 such that

k¢ can be obtained using 3¢ and 5¢ coins. ← P(k) inductive hypothesis

[We must show that P(k + 1) is true. That is:] We must show that

(k + 1)¢can be obtained using 3¢ and 5¢ coins. ← P(k + 1)

Case 1 (There is a 5¢ coin among those used to make up the k¢.): In this case replace the 5¢ coin by two 3¢ coins; the result will be (k + 1) ¢.

Case 2 (There is not a 5¢ coin among those used to make up the k¢.): In this case, because k ≥ 8, at least three 3¢ coins must have been used. So remove three 3¢ coins and replace them by two 5¢ coins; the result will be (k + 1) ¢.

Thus in either case (k + 1) ¢ can be obtained using 3¢ and 5¢ coins [as was to be shown].

[Since we have proved the basis step and the inductive step, we conclude that the proposition is true.]

ISS 225 CLASS NOTES VOCABULARY Abrogate - to abolish or do away with Court of Appeals - sit on panel of 3 Court Order - allows one party of a case to carry certain steps, this does not require probable cause Dicta - reasoning from a case that does not affect the holding, usually acknowledging a hypothetical situation that the court will not and/or does not need to respond to, if responded to it is in hopes of avoiding future cases En Banc - all of the justices on the court sit on the case EXCEPTION: 9th circuit because there are 29 justices, instead they sit in panels of 11 The Fruit of the Poisonous Tree Doctrine - states that if knowledge was illegally obtained, any legal information obtained based on the illegal information is unconstitutional and unjustly used because if other