3. Relationships Between Structures
“→” ≝ “Can be defined in terms of”
Programs
Groups
Proofs
Trees
Complex
numbers
Operators
Propositions
Graphs
Real numbers
Strings
Functions
Integers
As I mentioned earlier, mathematical structures can often be built up or defined in terms of simpler structures. This diagram (which itself is an example of the type of discrete structure known as a “graph”) outlines some ways in which a variety of discrete (and continuous) structures can ultimately be built up from a very simple type of discrete structure known as a set, which is one of the first structures we will learn. This is just to give you a a bit of a feel for the inter-relatedness of mathematical objects. This diagram is vastly simplified; some of the dependencies are left out, many other kinds of structures and ways of defining structures in terms of other structures are also possible and are left out for simplicity. For example, sets themselves can be defined in terms of functions, or relations. No one type of structure is truly fundamental, since almost any structure can be defined in terms of almost any other. Sets are only one possible starting point; but they are a popular one because their definition is so simple. We will revisit this diagram in more detail throughout this course, and show how the various links work.
Natural numbers
Matrices
Relations
Sequences
Infinite ordinals
Bits
n-tuples
Vectors
Sets
Not all possibilities are shown here.

4. Some Notations We’ll Learn

5. Why Study Discrete Math?
The basis of all digital information processing is: Discrete manipulations of discrete structures represented in memory.
It’s the basic language and conceptual foundation for all of computer science.
Discrete math concepts are also widely used throughout math, science, engineering, economics, biology, etc., …
A generally useful tool for rational thought!

6. Uses for Discrete Math in Computer Science
Advanced algorithms & data structures
Programming language compilers & interpreters.
Computer networks
Operating systems
Computer architecture
Database management systems
Cryptography
Error correction codes
Graphics & animation algorithms, game engines
The whole field!

7. Course Outline (as per Rosen)
Proof methods (§1.5)
Set theory (§1.6-7)
Functions (§1.8)
Number theory (§2.4-5)
Num. theory apps. (§2.6)
Matrices (§2.7)
Proof strategy (§3.1)
Sequences & sums (§3.2)
Logic (§1.1-3)
Inductive Proofs (§3.3)
Recursion (§3.4-5)
Combinatorics (ch. 4)
Basic Probability (§5.1)
Recurrences (§6.1-2)
More Counting (§6.5-6)
Relations (§7.1, -3, -5)
Graphs & trees (§8.1, 9.1)