4.4 Using Cases

Class: CSE 16 Subject: computer-science discrete-math Date: 2024-11-06 Teacher: Prof. Musacchio

Direct Proof

Introduction

• Sometimes we have to look at multiple cases before we can prove a statement is true

Example 1

• Proposition: If $n\in \mathbb{N}$, then $1+(-1{)}^n(2n-1)$ is a multiple of $4$.
  • Proof Suppose $n\in \mathbb{N}$(Then $n$ is either even or odd.)
  • Case 1
    • Suppose $n$ is even.
    • Then $n=2k$ for some $k\in \mathbb{Z}$, and $(-1)n=1$.
    • Thus $1+(-1)n(2n-1)=1+(1)(2\cdot 2k-1)=4k$, which is a multiple of $4$.
  • Case 2
    • Suppose n is odd.
    • Then $n=2k+1$ for some $k\in \mathbb{Z}$, and $(-1)n=-1$.
    • Thus $1+(-1)n(2n-1)=1-(2(2k+1)-1)=-4k$, which is a multiple of $4$.