MATH 302 Lecture 9

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Exam on Wednesday

Function Invertibility

A function $f$ is invertible if and only if it is bijective (both surjective/onto and injective/one-to-one)

Graphing A Function

Writing functions as columns and drawing arrows between them:

$\{(a,b)\in A\times B~|~f(a)=b\}$

Ceiling and Floor

Ceiling rounds up to next integer if decimal is greater than 0

Floor rounds down (truncates decimal)

Proposition. $\lfloor 2x\rfloor =\lfloor x\rfloor +\left\lfloor x+{\frac {1}{2}}\right\rfloor$

Proof. Let $x=n+\epsilon$ where $n\in \mathbb {Z}$ and $0\leq \epsilon <1$, thus $\lfloor x\rfloor =n$. There are two cases:

1. $0\leq \epsilon <{\frac {1}{2}}$
2. ${\frac {1}{2}}\leq \epsilon <1$

Proof of (1). $2x=2n+2\epsilon$, where $0\leq 2\epsilon <1$

$\lfloor 2x\rfloor =2n$

$\lfloor x\rfloor =\left\lfloor x+{\frac {1}{2}}\right\rfloor =n$

Proof of (2). $2x=2n+2\epsilon$

Review for Exam

1. Be familiar with definitions and terminology (key terms at end of each chapter 109-111)
2. Homework is fair game: logic proofs, set proofs, function proofs
3. Big-O, Big-Ω, Big-Θ
4. Algorithms
5. Quantifiers, properties and terminologies of conditionals
6. Membership Tables and set theory