수학적 귀납법 연습문제 — Sundstrom, Mathematical Reasoning · 4.2절 · 5번

Sundstrom, Mathematical Reasoning · 4.2절 · 5번영어 원문

Is the following proposition true or false? Justify your conclusion.

For each nonnegative integer

8^n \mid \left( {4n} \right)!

정답·풀이 보기 (영어 원문)

Let $P( n )$ be the predicate, “ $8^n \mid ( 4n )!$ .” Verify that $P( 0 ), P( 1 ), P( 2 )$ , and $P( 3 )$ are true. For the inductive step, the following fact about factorials may be useful:

$\begin{aligned}[4 (k + 1) ]! & = ( 4k + 4 )! \\ & = (4k + 4)(4k + 3)(4k + 2)(4k + 1) ( 4k )!\end{aligned}$

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.2절 · 12번영어 원문

Prove that for each natural number $n$ , any set with $n$ elements has $\dfrac{n \left( {n-1} \right)}{2}$ two-element subsets.

다 풀었으면 정답 확인

Let $P ( n )$ be, “Any set with $n$ elements has $\dfrac{n ( n - 1 )}{2}$ 2-element subsets.” $P ( 1 )$ is true since any set with only one element has no 2-element subsets. Let $k \in \mathbb{N}$ and assume that $P ( k )$ is true. This means that any set with $k$ elements has $\dfrac{k ( k - 1 )}{2}$ 2-element subsets. Let $A$ be a set with $k + 1$ elements, and let $x \in A$ . Now use the inductive hypothesis on the set $A - \left\{ x \right\}$ , and determine how the 2-element subsets of $A$ are related to the set $A - \left\{ x \right\}$ .

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