OpenCyber AcademyLesson map
Lesson 01 published. Map, example, EX001.
flowchart TD
N0["Learning outcomes"]
N1["1.1 Why precision matters"]
N2["1.2 Sets and membership"]
N3["1.3 Powers of two"]
N4["1.4 Counterexamples"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5
N5 --> EX["Worked example"]
EX --> Q["Exercise EX001"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["1.1 Why precision matters"]
N2["1.2 Sets and membership"]
N3["1.3 Powers of two"]
N4["1.4 Counterexamples"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> Q["EX001"]
[Learning outcomes]
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[1.1 Why precision matters]
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[1.2 Sets and membership]
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[1.3 Powers of two]
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[1.4 Counterexamples]
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[Synthesis]
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[Worked example] --> [Exercise EX001]
Learning outcomes
You will (1) use logic and sets to state network rules precisely; (2) reason with powers of two; (3) check claims with counterexamples.
1.1 Why precision matters
Priya Nair begins her bachelor's degree at the fictional Northbridge University and, in her first week, hears about the fictional Harbourside Community Clinics, four small clinics sharing one wireless passphrase for staff, visitors and medical carts. A tutor asks the class to write down, exactly, who should be able to reach what. Most first attempts are vague, such as "visitors should not see patient systems". Mathematical reasoning gives tools to make such statements precise enough to check, and precision is the first habit of network security, because a rule that can be read two ways will eventually be configured the wrong way.
1.2 Sets and membership
A set is a collection of distinct items, and much of network security is about membership. The set of clinic devices can be split into subsets: staff laptops, clinical carts, visitor phones and network equipment. A rule such as "no visitor device may reach any clinical device" becomes a statement about pairs drawn from two sets. Union, intersection and complement let Priya ask useful questions: which devices are in both the staff and clinical sets, which devices belong to no planned set, and whether the sets together cover every device seen on the network.
1.3 Powers of two
Computers count in binary, so network sizes come in powers of two. An address block with eight host bits holds two to the power eight, 256 addresses; seven bits hold 128; four bits hold 16. Each extra bit doubles the size, and each bit removed halves it. Priya practises moving between bit counts and sizes until it is automatic, because later she must size subnets for each clinic group. A common mistake is to treat sizes as decimal round numbers, asking for a block of 100 or 200, when only powers of two are possible.
1.4 Counterexamples
A general claim, such as "every visitor device is on the guest network", is disproved by a single counterexample: one visitor phone found on the staff network. Priya learns that security claims are usually universal, about all devices or all paths, so testing them means searching for counterexamples rather than collecting confirming cases. A hundred visitor phones correctly placed do not prove the claim; one misplaced phone disproves it. This idea returns throughout the course, in testing firewall rules, reviewing configurations and evaluating whether a control works.
Worked example
Priya writes the clinic rule as: for every device d in Visitors and every device c in Clinical, d cannot open a connection to c. She then lists what evidence would be a counterexample.
Synthetic case. The clinics and devices are synthetic.
Errors and limits
Informal language such as "usually" or "should" hides exceptions; precise statements make exceptions visible.
Exercises
- EX001 — Universal claim test. What single observation would disprove "no visitor device reaches clinical devices"?
Show marking criteria
Rubric: 3 = one visitor device successfully connecting to a clinical device, explained as a counterexample to a universal claim;; 2 = a visitor connecting;; 1 = mentions testing without a specific observation;; 0 = none. - EX002 — Bits to size. How many addresses does a block with six host bits contain?
Show marking criteria
Rubric: 3 = 64, from two to the power six, with the doubling rule explained;; 2 = 64;; 1 = a guess near 64 without method;; 0 = none. - EX003 — Unplanned devices. Using set language, describe devices that belong to no planned group.
Show marking criteria
Rubric: 3 = the complement of the union of planned sets, needing investigation;; 2 = devices outside all groups;; 1 = lists one device;; 0 = none.
Synthesis
Sets, powers of two and counterexamples turn vague security wishes into precise, checkable statements.