#discretemath — Public Fediverse posts
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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 set has `n` elements, its power set will always have `2^n` subsets!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Modulo Arithmetic:** The remainder after division.
Ex: `10 mod 3 = 1` (since `10 = 3*3 + 1`).
Pro-Tip: Super useful for clocks & cyclical patterns – it wraps around! -
Alright, future engineers!
**Pigeonhole Principle:** If `n` items go into `m` containers & `n > m`, at least one container has >1 item.
Ex: 7 pigeons, 6 holes => one hole has >= 2 pigeons.
Pro-Tip: Great for proving existence without finding *which* one!
#DiscreteMath #Proof #STEM #StudyNotes -
Alright, future engineers!
**Combinations:** Ways to choose items from a set where order *doesn't* matter.
Ex: Picking 3 ice cream flavors from 10: C(10,3) = 120 ways.
Pro-Tip: Think teams or groups – the order of selection is irrelevant!
#Combinatorics #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Power Set:** The set of all subsets of a given set, including the empty set & the set itself.
Ex: For `S={1,2}`, `P(S) = {{}, {1}, {2}, {1,2}}`.
Pro-Tip: Its size is `2^|S|`! A quick check for completeness.
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Combinations:** Ways to choose items from a set where order DOES NOT matter.
Ex: Choosing 3 students from 5 for a team: `C(5,3) = 10` ways.
Pro-Tip: Order is IRRELEVANT! Think committees, not lineups.
#Combinatorics #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Modular Arithmetic:** Numbers wrap around after a certain value (the modulus).
Ex: `(10 + 4) mod 12 = 14 mod 12 = 2`. Think of a clock!
Pro-Tip: It's super useful in cryptography & computer science for cycles/remainders!
#NumberTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Permutation:** Ways to arrange 'k' items from 'n' where order *matters*.
Ex: P(n,k) = n!/(n-k)!. Arrange 3 from 5: P(5,3) = 60.
Pro-Tip: Think 'P' for position – order matters! -
Alright, future engineers!
**Cartesian Product:** All ordered pairs (a,b) where `a` is from Set A & `b` is from Set B.
Ex: A={1,2}, B={x,y} -> AxB = {(1,x), (1,y), (2,x), (2,y)}.
Pro-Tip: If |A|=m, |B|=n, then |AxB|=m*n! (Total possibilities!)
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Modulo Arithmetic:** Finding the remainder after division.
Ex: `17 mod 5 = 2`.
Pro-Tip: Crucial for cryptography, clock arithmetic, and hashing!
#DiscreteMath #NumberTheory #STEM #StudyNotes -
Alright, future engineers!
**Power Set:** The set of all subsets of a set, including empty & the set itself.
Ex: S={A,B}, P(S) = {{}, {A}, {B}, {A,B}}.
Pro-Tip: A set with 'n' elements has `2^n` subsets. Essential for understanding relations!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Modular Arithmetic:** Numbers wrap around after reaching a specific value (the modulus).
Ex: `10 mod 3 = 1` (since 10 = 3*3 + 1).
Pro-Tip: Think of a clock! Useful for cryptography & algorithms.
#DiscreteMath #NumberTheory #STEM #StudyNotes -
Alright, future engineers!
**Set Union (A U B):** The set containing all elements that are in set A, or in set B, or in both.
Ex: A={1,2}, B={2,3} => A U B = {1,2,3}.
Pro-Tip: Think 'OR' – an element is in the union if it's in A *OR* in B. Visualize with Venn Diagrams! -
Alright, future engineers!
**Combinations:** The number of ways to choose items from a set where *order does NOT matter*.
Ex: Choosing 3 members from 5 for a team: C(5,3) = 10.
Pro-Tip: If re-ordering chosen items doesn't make a new group, it's a combination!
#Combinatorics #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Permutations:** The number of ways to arrange items from a set where *order matters*.
Ex: Arranging 3 books from 5 distinct books: P(5,3) = 60.
Pro-Tip: If changing the order gives a *new* distinct arrangement, it's a permutation!
#Combinatorics #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Big O Notation:** Describes an algorithm's worst-case performance as input size 'n' grows.
Ex: `O(n^2)` for nested loops.
Pro-Tip: Essential for picking scalable algorithms! Smaller 'O' means better performance for large datasets.
#DiscreteMath #Algorithms #STEM #StudyNotes -
Alright, future engineers!
**Cartesian Product:** A set of all possible ordered pairs where the first element is from the first set & the second from the second.
Ex: A={1,2}, B={a,b} -> AxB = {(1,a), (1,b), (2,a), (2,b)}
Pro-Tip: `|AxB| = |A| * |B|`. Essential for combining domains!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Modulo Arithmetic:** The remainder when an integer is divided by another.
Ex: `17 mod 5 = 2` (17 / 5 leaves 2).
Pro-Tip: Think clock arithmetic! Crucial for cryptography, hashing, and cyclic systems.
#DiscreteMath #NumberTheory #STEM #StudyNotes -
Alright, future engineers!
**Graph:** A set of nodes (vertices) linked by connections (edges).
Ex: Friends in a social network are nodes, friendships are edges.
Pro-Tip: Use graphs to model relationships & find optimal paths/flows.
#GraphTheory #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 will always have 2^n elements. Count 'em!
#SetTheory #DiscreteMath #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!
**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 -
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! -
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