Agenda, Semester Week 2

Agenda, Semester Week 2#

Sets and Functions#

This week briefly introduces the foundations of set theory and then continues with a focus on general proporties of functions. We will study the inverse functions and compositions of functions, and will pay special attention to so-called recursively defined functions. A particularly important category of functions that we will study is the trigonometric functions and their inverses.

Key Terms#

Long Day will cover:

  • Sets, set notation and set relations.

  • Set operations: union \(\cup\), intersection \(\cap\), difference \(\setminus\), Cartesian product \(\times\).

  • The classical sets of numbers: \(\,\mathbb N,\, \mathbb Z,\, \mathbb Q,\, \mathbb R\,\).

  • Functions and functional expressions.

  • The domain, co-domain and image set/range.

  • Composite functions \(\circ\) and the identity function \(\mathrm{id}\).

Short Day will cover:

  • Inverse functions.

  • Piecewise defined functions and recursively defined functions.

  • Summation notation \(\sum\).

Preparation and Syllabus#

Syllabus this week is Chapter 2, excluding the subsections on trigonometric functions and their inverses, and the first two sections of Chapter 3.

  • For Long Day, read Chapter 2 through Example 2.3.2.

  • For Short Day, read Sections 3.1 and 3.2.

Exercises#

Exercises for Long Day.

Exercises for Short Day.

Contents