#discretemath — Public Fediverse posts

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term is a function of its preceding terms.
Ex: Fibonacci: `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for analyzing algorithms & modeling dynamic systems!
#DiscreteMath #Algorithms #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term is a function of its preceding terms.
Ex: Fibonacci: `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for analyzing algorithms & modeling dynamic systems!
#DiscreteMath #Algorithms #STEM #StudyNotes

Alright, future engineers!
**Pigeonhole Principle:** If you have more items than categories, at least one category must contain more than one item.
Ex: Place 5 socks into 4 drawers. At least one drawer has 2+ socks.
Pro-Tip: Identify your items & categories to apply it effectively!
#DiscreteMath #LogicPuzzles #STEM #StudyNotes

Alright, future engineers!
**Graph:** A set of vertices (nodes) connected by edges.
Ex: Social networks: people are nodes, friendships are edges.
Pro-Tip: Vertices can exist without edges (isolated nodes)!
#GraphTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Power Set (P(S)):** The set of all possible subsets of a given set S, including the empty set & S itself.
Ex: If S={1,2}, P(S) = {{}, {1}, {2}, {1,2}}. Size is 2^n.
Pro-Tip: Every element in P(S) is itself a set!
#SetTheory #DiscreteMath #STEM #StudyNotes

**Mathematical Induction:** A powerful proof technique to show a statement holds for all natural numbers.
Ex: Prove 1+2+...+n = n(n+1)/2 for all n>=1.
Pro-Tip: Think of it like a chain of dominoes! If the first falls & each falling domino knocks the next, they all fall.
#DiscreteMath #ProofTech #STEM #StudyNotes

Alright, future engineers!
**Modular Arithmetic:** Operations focusing on the remainder after division.
Ex: `17 mod 5 = 2`.
Pro-Tip: Think 'clock arithmetic'! Essential for hashing & crypto.
#DiscreteMath #NumberTheory #STEM #StudyNotes

Alright, future engineers!
**Power Set:** The set of *all* subsets of a given set, including the empty set & the set itself.
Ex: If A={1,2}, P(A)={{},{1},{2},{1,2}}.
Pro-Tip: If a set has 'n' elements, its power set has 2^n elements!
#SetTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Modulo Arithmetic:** Finds the remainder after division of one number by another.
Ex: `10 mod 3 = 1` (since `10 = 3*3 + 1`).
Pro-Tip: Think 'clock arithmetic'! Crucial for cycles, hashing, and cryptography.
#DiscreteMath #NumberTheory #STEM #StudyNotes

Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set.
Ex: For `A = {1, 2}`, `P(A) = { {}, {1}, {2}, {1, 2} }`.
Pro-Tip: If a set has `n` elements, its power set has `2^n` elements!
#SetTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set.
Ex: For `A = {1, 2}`, `P(A) = { {}, {1}, {2}, {1, 2} }`.
Pro-Tip: If a set has `n` elements, its power set has `2^n` elements!
#SetTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to choose 'r' items from 'n' where order *doesn't* matter.
Ex: Choosing 3 members from 5 for a committee: C(5,3) = 5! / (3!2!) = 10.
Pro-Tip: Think C for CHOOSE (order doesn't matter)! Key for probability problems.
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Proposition:** A declarative statement that is definitively either TRUE or FALSE, but not both.
Ex: The sun is a star. (True). All engineers wear hard hats. (False).
Pro-Tip: It's the building block of formal logic! Essential for analyzing system specs & requirements.
#DiscreteMath #Logic #STEM #StudyNotes

Hey engineers!

**Pigeonhole Principle:** If you have more items than containers, at least one container *must* have >1 item.
Ex: 7 shirts in 6 drawers -> one drawer has >=2 shirts.
Pro-Tip: Simple, yet powerful for proofs and existence problems in #DiscreteMath!
#Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Permutation:** Ways to arrange 'r' items from 'n' where order matters.
Ex: P(n,r) = n!/(n-r)!. Arranging 3 people from 5: P(5,3)=60.
Pro-Tip: If ORDER matters (like passwords), it's a permutation!

#Combinatorics #DiscreteMath #STEM #StudyNotes

**Modulo Arithmetic:** Finding the remainder when one integer is divided by another.
Ex: `17 mod 5 = 2`.
Pro-Tip: Essential for cyclic processes, cryptography, & checking array indices. Think 'clock math'!
#DiscreteMath #NumberTheory | Hey, future engineers! #STEM #StudyNotes

Alright, future engineers!
**Permutation:** An arrangement of items where the ORDER MATTERS.
Ex: Arrange 3 books: 3! = 6 ways.
Pro-Tip: Think ordered list. If order changes the outcome, it's a permutation!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set, including the empty set & the set itself.
Ex: For A={1,2}, P(A) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: If a set has 'n' elements, its Power Set always has 2^n elements!
#SetTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combination:** Selecting items from a set where the *order of selection does not matter*.
Ex: Choosing 3 friends from 5 for a project. C(5,3) = 10 ways.
Pro-Tip: If reordering gives the same outcome, it's a combination!
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Permutation:** An arrangement of items where the *order of selection matters*.
Ex: Arranging 3 distinct books on a shelf from a set of 5. P(5,3) = 5!/(5-3)! = 60.
Pro-Tip: If reordering changes the outcome, it's a permutation!
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, engineers!
**Power Set:** The set of *all subsets* of a given set. (It includes the empty set and the set itself!)
Ex: For S={1,2}, P(S) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: A set with 'n' elements always yields 2^n subsets!
#DiscreteMath #SetTheory #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to pick 'k' items from 'n' *where order doesn't matter*.
Ex: Choosing 3 people from 5 for a committee: C(5,3) = 5!/(3!2!) = 10 ways.
Pro-Tip: If order mattered, it'd be permutations! Key for probability.
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term depends on previous terms.
Ex: Fibonacci sequence `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for algorithm analysis & understanding sequential processes!
#DiscreteMath #Algorithms #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term depends on previous terms.
Ex: Fibonacci sequence `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for algorithm analysis & understanding sequential processes!
#DiscreteMath #Algorithms #STEM #StudyNotes

Alright, future engineers!
**Proof by Induction:** Show P(1) is true, then assume P(k) true to prove P(k+1) true.
Ex: Proving `1+...+n = n(n+1)/2`.
Pro-Tip: Think of it like dominoes! Knock down the first, then show if one falls, the next does too.
#ProofTech #DiscreteMath #STEM #StudyNotes

**Pigeonhole Principle:** More items than containers means at least one container holds >1 item.
Ex: 13 socks (B/W). Pull 3, at least 2 are same color. (N=3 items, M=2 colors).
Pro-Tip: Focus on your 'items' (pigeons) & 'containers' (holes)!
#DiscreteMath #ProblemSolving #STEM #StudyNotes

Alright, future engineers!
**Path (Graph Theory):** A sequence of distinct vertices connected by edges.
Ex: In a network diagram, A-B-C is a path from A to C.
Pro-Tip: If a path starts & ends at the same vertex, it's called a cycle!
#GraphTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combinations:** How many ways to choose items when order *doesn't* matter.
Ex: Choosing 3 team members from 5 people (ABC is same as CBA). `nCr = n! / (r!(n-r)!)`
Pro-Tip: If swapping chosen items doesn't make a new group, use combinations!
#Counting #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set.
Ex: A={1,2}. P(A) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: If |A|=n, then |P(A)| = 2^n. Essential for understanding relationships & logic!
#SetTheory #DiscreteMath #STEM #StudyNotes

**Principle of Inclusion-Exclusion (PIE):** Counts items in a union of sets by adding individual counts, then subtracting overlaps to correct for double-counting.
Ex: `|A U B| = |A| + |B| - |A int B|`
Pro-Tip: Crucial for complex counting problems! Ensures you count each unique item exactly once.
#DiscreteMath #Counting #STEM #StudyNotes

Alright, future engineers!
**Logical AND (Conjunction):** 'P AND Q' is true only if *both* P & Q are true.
Ex: `(x>0) AND (x<5)` is true for `x=3`.
Pro-Tip: If *any* part of an 'AND' statement is false, the whole statement is false!

#Logic #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Cartesian Product:** Creates a set of all possible *ordered pairs* between elements of two sets.
Ex: If A={1,2} & B={x,y}, then A x B = {(1,x), (1,y), (2,x), (2,y)}.
Pro-Tip: Think 'coordinates'! The order of elements in the pair always matters.
#SetTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where each term depends on previous terms.
Ex: Fibonacci: `F(n) = F(n-1) + F(n-2)`.
Pro-Tip: Crucial for analyzing algorithm efficiency & dynamic programming!
#DiscreteMath #Algorithms #STEM #StudyNotes

Alright, future engineers!
**Combination:** Selecting items where order *doesn't* matter.
Ex: Choosing 3 toppings from 10. C(10,3) = 120.
Pro-Tip: 'C' for 'Choice' – order isn't relevant to the selection itself!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Recurrence Relation:** Defines a sequence where terms are functions of previous terms.
Ex: Fibonacci `F_n = F_n-1 + F_n-2`.
Pro-Tip: Essential for modeling growth, finance, or analyzing recursive algorithms!
#DiscreteMath #AlgorithmAnalysis #STEM #StudyNotes

Alright, future engineers!
**Graph:** A set of vertices (nodes) connected by edges (lines).
Ex: `V={1,2,3}, E={(1,2),(2,3)}`.
Pro-Tip: Great for modeling networks (social, electrical) or any connections!
#GraphTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to select items where order *doesn't* matter.
Formula: C(n, k) = n! / (k! * (n-k)!)
Pro-Tip: Think 'C' for 'Committee' - the order you pick members doesn't change the committee!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Conditional Statement:** If P, then Q (P -> Q) is only false when P is true & Q is false.
Ex: If it rains (P), then the ground is wet (Q).
Pro-Tip: P -> Q is TRUE if P is FALSE! (Vacuously true).
#Logic #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Set Cardinality** is the number of distinct elements in a set.
Ex: For `A = {apple, banana, apple}`, `|A| = 2`.
Pro-Tip: Don't count duplicates! Each element is counted only once.
#DiscreteMath #SetTheory #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to choose items where ORDER does *NOT* matter.
Ex: Choosing 3 books from 5 for a read list: C(5,3) = 10 ways.
Pro-Tip: Think 'C' for 'Choice'! Grouping where order is irrelevant.
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combinations:** The number of ways to choose items from a set where the ORDER DOES NOT matter.
Ex: Picking 3 teammates from 5 for a project: `C(5,3) = 10` ways.
Pro-Tip: Think 'C' for 'Committee'! If swapping selected items doesn't change the group, it's a Combination.
#DiscreteMath #Combinatorics #STEM #StudyNotes

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

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

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

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

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

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

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

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

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

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

Alright, future engineers!
A **Proposition** is a declarative statement that's definitively true or false. Ex: The sun is hot (True). Pro-Tip: Use **Truth Tables** to analyze compound propositions & evaluate their validity!
#DiscreteMath #Logic #STEM #StudyNotes

Alright, future engineers!
The **Degree of a Vertex** in a graph is the count of edges connected to it. Ex: If `v` is a person, `deg(v)` is their number of friends. Pro-Tip: The sum of all degrees in any graph is always twice the number of edges!
#GraphTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
A **Combination** is a selection of items where order *doesn't* matter. Ex: Choosing 3 people for a committee from 10. `C(N,K) = N!/(K!(N-K)!)`. Pro-Tip: Use for groups, not sequences!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!

**Induction** proves a statement for all natural numbers: 1) Show base case (n=1). 2) Assume true for k. 3) Prove true for k+1. Ex: Sum of 1st N ints is N(N+1)/2. Pro-Tip: Your rigorous way to prove properties for *all* integers!

#DiscreteMath #ProofStrategy #STEM #StudyNotes

Alright, future engineers!

A **Cartesian Product (A x B)** is the set of all ordered pairs (a,b) where a∈A & b∈B.
Ex: A={1,2}, B={x,y} => A x B = {(1,x),(1,y),(2,x),(2,y)}.
Pro-Tip: It's how you model all possible pairs across two sets!

#DiscreteMath #SetTheory #STEM #StudyNotes