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Past exam of the year 1 of the computer science course of the University of Oxford 2016 Discrete mathematics, continuous mathematics and probability (resit) 2 a Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31A binary relation on is an equivalence relation exactly when it is reflexive, symmetric, and transitive:The equivalence class of is . The equivalence classes partition : every element belongs to its own class, and two classes are either equal or disjoint.
Past exam of the year 1 of the computer science course of the University of Oxford 2016 Discrete mathematics, continuous mathematics and probability (resit) 1 c Solution by Codex 0 2026-10-31
Codex 0 2026-10-31A function is onto, or a surjective function, when every has a preimage: . It is one-to-one, or an injective function, when implies for all . It is bijective, a bijection, when both properties hold.
An inverse function is a function satisfyingFor a bijection, is the unique with : existence follows from surjectivity and uniqueness from injectivity. This is a two-sided inverse function, rather than just a left or right inverse.
Past exam of the year 1 of the computer science course of the University of Oxford 2016 Discrete mathematics, continuous mathematics and probability (resit) 1 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31By definition the symmetric difference is . Put and . Distribution of set intersection over set union givesFor disjoint sets, set difference removes nothing, so . ConsequentlyHere the second equality combines with ; the last uses .
Past exam of the year 1 of the computer science course of the University of Oxford 2016 Digital systems, linear algebra, introduction to formal proof 8 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Write . Use negation introduction on the existential claim. For its fresh witness , instantiate ; the resulting self-application says both and its reverse. The given law of excluded middle then provides the two cases.Both case assumptions are discharged at line 15; the witness assumption is discharged at line 16. Since is absent from the surrounding premise and the exported conclusion , existential quantifier elimination rule is valid. Line 17 is the required negated existential. This is the diagonal self-application underlying the barber paradox.
1 ∀x. ∀y. (S(x,y) ∨ ¬S(x,y)) premise
2 | ∃b. H(b) assumption
3 | | H(b) assumption, b fresh
4 | | (S(b,b) → ¬S(b,b)) ∧ (¬S(b,b) → S(b,b)) ∀-elim 3, p := b
5 | | S(b,b) → ¬S(b,b) ∧-elim-L 4
6 | | ¬S(b,b) → S(b,b) ∧-elim-R 4
7 | | ∀y. (S(b,y) ∨ ¬S(b,y)) ∀-elim 1, x := b
8 | | S(b,b) ∨ ¬S(b,b) ∀-elim 7, y := b
9 | | | S(b,b) assumption
10 | | | ¬S(b,b) →-elim 5,9
11 | | | ⊥ ¬-elim 9,10
12 | | | ¬S(b,b) assumption (new case)
13 | | | S(b,b) →-elim 6,12
14 | | | ⊥ ¬-elim 13,12
15 | | ⊥ ∨-elim 8,9–11,12–14
16 | ⊥ ∃-elim 2,3–15
17 ¬∃b. H(b) ¬-intro 2–16 Past exam of the year 1 of the computer science course of the University of Oxford 2016 Digital systems, linear algebra, introduction to formal proof 7 c Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Proof by contradiction, or reductio ad absurdum, has the formdischarging the assumption . This is the classical rule that concludes directly, as opposed to the negation introduction rule, which concludes a negation.
To derive double-negation elimination using RAA without assuming double-negation elimination itself, useHere line 1 is the negation of line 2. Thus only negation elimination rule inside the subproof, followed by RAA, is needed.
1 ¬¬φ premise
2 | ¬φ assumption
3 | ⊥ ¬-elim 2,1
4 φ RAA 2–3 Past exam of the year 1 of the computer science course of the University of Oxford 2016 Digital systems, linear algebra, introduction to formal proof 7 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Let . In the following natural deduction proofs, indentation and vertical bars mark assumption scopes. Rule names are the introduction and elimination rules supplied by the paper.
First work backwards from the premise using disjunction elimination, proving the whole desired logical conjunction in each case:Alternatively, first transform the goal by conjunction introduction, obtaining and by separate disjunction elimination proofs:The essential difference is the final rule: the first proof splits the premise once, while the second splits the goal first and then analyzes the premise twice. Counting only rule applications, both displayed proofs take nine inference steps. The first uses four conjunction elimination steps, two disjunction introduction steps, two conjunction introduction steps and one disjunction elimination; the second replaces one conjunction introduction with a second disjunction elimination. If assumption lines are counted as displayed proof lines, the second is longer, 14 rather than 12, because it opens four case assumptions rather than two. Thus a raw line count and an inference-step count need not agree.
1 H premise
2 | P ∧ Q assumption
3 | P ∧-elim-L 2
4 | Q ∧-elim-R 2
5 | Q ∨ R ∨-intro-L 4
6 | P ∧ (Q ∨ R) ∧-intro 3,5
7 | P ∧ R assumption (new case)
8 | P ∧-elim-L 7
9 | R ∧-elim-R 7
10 | Q ∨ R ∨-intro-R 9
11 | P ∧ (Q ∨ R) ∧-intro 8,10
12 P ∧ (Q ∨ R) ∨-elim 1,2–6,7–111 H premise
2 | P ∧ Q assumption
3 | P ∧-elim-L 2
4 | P ∧ R assumption (new case)
5 | P ∧-elim-L 4
6 P ∨-elim 1,2–3,4–5
7 | P ∧ Q assumption
8 | Q ∧-elim-R 7
9 | Q ∨ R ∨-intro-L 8
10 | P ∧ R assumption (new case)
11 | R ∧-elim-R 10
12 | Q ∨ R ∨-intro-R 11
13 Q ∨ R ∨-elim 1,7–9,10–12
14 P ∧ (Q ∨ R) ∧-intro 6,13 Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
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Further documentation can be found at: docs.ourbigbook.com
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