Crap.

Crap, crap, crap, crap!

Lexus cursed under his breath as he scrambled to fish his paper, pen and laptop out of his bag and threw them onto his desk. He crashed onto his swivel chair and pulled the desk over him. Using one hand to unlock his laptop and the other to unlock his phone, Lexus pulled up both the lecture notes that Arthur had sent him many days ago as well as the problem sheet that was set for that lecture.

Lexus glanced over at the clock and felt his heart quicken. Problem sheets for the discrete mathematics course were due 5pm the day before a supervision, and it was currently 2pm. In other words, he had 3 hours to catch up on the lecture he missed, complete the - he scrolled through to check - 13 questions on the problem sheet, then scan and submit it to his supervisor, Professor Emerson.

He could blame the system for missing his lecture, but he had no one to blame but himself for missing the lecture then forgetting catch up and to do the problem sheet.

The first half hour was spent looking through Arthur's lecture notes and searching up anything that he didn't quite understand. By the time he understood all about the equivalence relations and set partitions , there were only two and a half hours left on the clock.

The problem sheet was split into two sections: 6 basic exercises, and 7 core exercises, which were one difficulty level higher. There was also the optional advanced exercises which Lexus would normally attempt, but he didn't have time for that.

First question proving that ⊆ is a partial order. Trivial. Lexus sped through the question, proving its reflexive, transitive and antisymmetric properties.

The second, third and fourth questions were of similar style and difficulty, mostly about finding this or that of a specific set, or proving simple statements. He didn't find any of them particularly hard, and managed to complete the first four questions in twenty minutes.

At this point, Lexus felt pretty confident. He had over two hours left, and at this pace, he would finish the problem sheet with time to spare.

Question 5 was a tedious manoeuvre of listing out sets and finding their unions and intersections. The question wasn't difficult, but Lexus spent a good ten minutes just writing out the numbers.

Two hours left for eight questions. Fifteen minutes per question, easy.

Question 6 was another proof question with three parts. The first two parts were relatively simple, proving mutual exclusivity and double complement elimination, but the last part, proving De Morgan's Laws, was less so. Fortunately, he had seen the proof before while reading "A First Course on Probability", and understood the general principles of the proof well enough to reproduce it from memory.

Lexus scribbled down:

Proving De Morgan’s First Law (A ∪ B)′ = A′ ∩ B′:

Let set R = (A ∪ B)′ and set S = A′ ∩ B′

Let y be an element in set R, i.e. y ∈ R

Then y ∈ (A ∪ B)′

⇒ y ∉ A ∪ B

⇒ y∉ A and y∉ B

⇒ y ∈ A′ and y ∈ B′

⇒ y ∈ A′ ∩ B′

⇒ y ∈ S

Therefore R ⊆ S. Reversing the argument, we can show that S ⊆ R. Hence S = R, proving De Morgan’s First Law.

Proving De Morgan’s Second Law (A ∩ B)′ = A′ ∪ B′

Let set R = (A ∩ B)′ and set S = A′ ∪ B′

Let y be an element such that y ∈ R

Then y ∈ (A ∩ B)′

⇒ y∉ A ∩ B

⇒ y ∉ A or y ∉ B

⇒ y ∈ A′ or y ∈ B′

⇒ y ∈ A′ ∪ B′

⇒ y ∈ S

Therefore R ⊆ S. Reversing the argument, we can show that S ⊆ R. Hence S = R, proving De Morgan’s Second Law.

He took some corners with showing the converse argument that S ⊆ R, but it would have to do.

Another glance at the time. One hour and forty five minutes left. Onto the core exercises.

Lexus steadily worked through each of the questions one by one, making sure to keep track of the time. He worked through it in the same way as he would in an exam, skipping over the parts he got stuck on, hoping that he would have enough time to come back to it before the deadline. He had briefly considered asking for an extension, but quickly dismissed the idea. Professor Emerson was a strict man, and Lexus was going to get slaughtered if his only reason was "I forgot".

When Lexus arrived at the final question, he had precisely twenty minutes left. Time was tight. He still had to scan the work and send it over to the professor. Lexus clutched his pen even tighter and willed his brain to work at full pace.

For a family F ⊆ P(A), let T ≜ {U ⊆ A | ∀S ∈ F. S ⊆ U }. Prove that ∪ F = ∩ T.

The first thing that Lexus had to do was to wrap his head around all the symbols.

A family is a set of sets, and P(A) referred to the set of all possible subsets of A, including A itself and the empty set. The ≜ symbol refers to a definition, so T is defined to be the set of all Us. U, on the other hand, is the set of all possible subsets of A such that for every element S which is an element of F, S is a subset of U. And Lexus had to show that the union of all the sets in F is equal to the intersection of all the sets in T.

Great.

Lexus focused his mind, and got to work.

It was 4:56pm when Lexus wrote down the final symbol. He quickly grabbed his phone, opened the scanner app and took pictures of the eight pages of working from the past two hours of work. Exporting to pdf...Completed!

Lexus quickly drafted the email, attached the document, and sent it off to Professor Emerson. After pressing the send button, he checked the timestamp of the email. 4:58pm. Phew.

He crashed back onto his chair and sat there for a good ten minutes, contemplating his life decisions.

=====

After submitting the piece of work to Professor Emerson, Lexus spent the rest of the evening reviewing the past lecture, attempting the optional questions and consolidating his knowledge about sets and their properties. At Cambridge, you couldn't afford to go into a supervision without prior preparation, not if you wanted to come out in a single piece. There were only two people, him and Arthur, facing Professor Emerson, and it would be incredibly obvious if Lexus didn't put in enough effort.

---

Mathematics EXP +1

---

Lexus metaphorically rolled his eyes. "System, why didn't you give me any experience when I was working my fingers to the bone trying to finish my problem sheet?"

"You know better than anyone why you didn't get any experience," the system said. "Could you really call that learning?"

"Well...maybe?"

Lexus heard a scoff.

=====

The next morning, Lexus walked towards one of the many circular wooden tables on the ground floor of the college, which oversaw a vast pasture of green. He gave a slight wave to Arthur, who was already sitting there, and took a seat next to him. A few minutes later, Professor Emerson appeared and the supervision officially started.

"So, how did the two of you find the problem sheet for this week?" Professor Emerson asked.

Lexus didn't want to betray the fact that he only realised he had a problem sheet to do three hours before the deadline, so he let Arthur do the talking. "It was generally okay", Arthur said.

Professor Emerson wasn't going to let Lexus off the hook, though. "Lexus, and how was it for you?"

Recalling the system's words, Lexus knew that Professor Emerson, who was a distinguished professor in the field of theoretical computer science, would probably be at level 7 or 8 according to the system's standards. There was no way a meagre level 0 like Lexus would be able to escape Professor Emerson's discerning eyes, so Lexus decided to be honest.

"Most of the questions were quite straightforward, but I didn't manage to give it my best shot."

Professor Emerson nodded. "I could tell that you were rushing things. It definitely wasn't your best work, with all the cut corners and lack of explanation. Despite the subpar effort though, your foundations in this topic are solid, so I'm not too worried."

"Yeah, Lexus had been working like mad the past few days," Arthur chipped in.

Lexus couldn't help but smile at the sudden compliment. He had been so preoccupied with reading the books and completing the quest that he had neglected his other work. He had expected to have been reprimanded for falling behind, but the professor said that his foundations were good, so maybe he was smarter than he initially thought?

As if reading his mind, the system spoke. "It's simply because the chapter you were reading on *A First Course on Probability* was coincidentally on sets and their properties. Had it been any different, you would have been screwed."

Hand it to the system to bring his ego back to earth.

The rest of the supervision went on uneventfully. As Lexus had taken the time the previous evening to go over the content again, he was relatively prepared and could ask good questions as well as answer them. Getting taught by a level 7 or 8 professor was always a valuable experience, and Lexus took away a lot from the session. Not just in terms of maths, but in terms of points. By the end of the hour of supervision, Lexus had gained a total of 5 experience points, many more than he could have achieved by himself in such a short period of time.

After the supervision ended, Lexus and Arthur went out to eat lunch before heading back to their respective rooms to grab the things they need to study in the library.

As soon as Lexus entered his room, he whispered, "System, status!" and a screen appeared in front of him.

---

Name: Lexus Kagan

Current Quest: Background Knowledge [1/10]

Computer Science: Level 0 [EXP 5/100 (5%)]

Mathematics: Level 0 [EXP 36/100 (36%)]

Physics: Level 0 [EXP 0/100 (0%)]

Engineering: Level 0 [EXP 0/100 (0%)]

Chemistry: Level 0 [EXP 0/100 (0%)]

Biology: Level 0 [EXP 0/100 (0%)]

---

Counting back, it had only been five days since the system whooshed into his life, and Lexus was already over a third of the way there to level 1. He had already found enough value in the system even at his current level, but he couldn't wait to find out what would happen once he levelled up.

Would answers magically come to him when he looks a question? Would he be able to ace all his exams? Would he suddenly be the best in his year? Would he become a legend, a prodigy?

The curiosity was eating away at his patience. Lexus threw the things he needed in his bag, and immediately marched off to meet Arthur at the college library.