Mastodawn

Alright, future engineers!

**Modular Arithmetic:** `a === b (mod n)` means `a` & `b` have the same remainder when divided by `n`.
Ex: `17 === 5 (mod 12)` since `17 % 12 = 5` & `5 % 12 = 5`.
Pro-Tip: Think 'clock arithmetic'! It's key for cryptography & hashing.

#NumberTheory #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 28

Alright, future engineers!
**Pigeonhole Principle:** If `n` items are put into `m` containers, and `n > m`, then at least one container must contain more than one item.
Ex: 7 emails into 6 folders means one folder has >1 email.
Pro-Tip: A powerful proof technique for existence!
#DiscreteMath #Logic #STEM #StudyNotes

kipriyanovich Apr 27

Alright, future engineers!
**Combinations:** Ways to *choose* items from a set where the order *doesn't* matter.
Ex: Picking 3 teammates from 5 friends: `C(5,3) = 10` ways.
Pro-Tip: If swapping items doesn't create a new outcome, it's a combination!
#DiscreteMath #Combinatorics #STEM #StudyNotes

kipriyanovich Apr 27

Alright, future engineers!
**Probability Mass Function (PMF)**: Gives the probability for *each specific outcome* of a discrete random variable.
Ex: Fair die `P(X=3) = 1/6`.
Pro-Tip: All `P(X=x)` values must be non-negative & sum to 1.
#Probability #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 26

Alright, future engineers!
**Modulo Arithmetic** (a mod n) finds the remainder when integer 'a' is divided by integer 'n'.
Ex: 7 mod 3 = 1 (since 7 = 2*3 + 1).
Pro-Tip: Think of clocks! 13:00 is 1:00 (13 mod 12 = 1). Essential for cryptography!
#NumberTheory #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 24

Alright, future engineers!
The **Union** of sets A & B (A U B) is all elements in A *or* B.
Ex: A={1,2}, B={2,3}. A U B = {1,2,3}.
Pro-Tip: Think 'OR' for Union. It includes everything from either set!
#SetTheory #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 24

Alright, future engineers!

**Combinations**: ways to choose items where order *doesn't* matter.
Ex: Picking 3 committee members from 5 people: C(5,3) = 10 ways.
Pro-Tip: Think 'selecting ingredients for a soup' – the order you add them doesn't change the final soup!

#Combinatorics #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 23

Future engineers!
An **Adjacency Matrix** (A) shows graph connections: A[i][j]=1 if vertex i links to j, else 0.
Ex: For a-b, b-c: A[a,b]=1, A[b,a]=1.
Pro-Tip: It's symmetric for undirected graphs!
#GraphTheory #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 18

Alright, future engineers!
**Permutations** count arrangements where order *matters*.
Ex: Electing a Pres, VP, Sec from 10 people: P(10,3) = 10!/(10-3)!
Pro-Tip: If switching item order creates a *new* distinct outcome, it's a permutation!
#Probability #DiscreteMath #STEM #StudyNotes

kipriyanovich Apr 15

Alright, future engineers!
**Pigeonhole Principle:** If you have more items than containers, at least one container must contain more than one item. Ex: 5 emails to 4 folders = at least 1 folder gets 2+ emails. Pro-Tip: Think worst-case distribution to guarantee the outcome!
#DiscreteMath #Logic #STEM #StudyNotes