Alright, future engineers!
**Graph:** A collection of vertices (nodes) connected by edges (links).
Ex: A social network (people=vertices, friendships=edges).
Pro-Tip: Visualize connections & relationships! Essential for network analysis & system design.
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**Pigeonhole Principle:** If `n` items are put into `m` containers with `n > m`, at least one container has >1 item.
Ex: 3 socks in 2 drawers -> one drawer has at least 2 socks.
Pro-Tip: Count items (pigeons) & categories (holes) to prove existence!
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Alright, future engineers!
**Mathematical Induction:** A powerful proof method to show a statement holds for *all* natural numbers.
Ex: Prove 1+2+...+n = n(n+1)/2.
Pro-Tip: It's like setting up dominoes! Show the first falls, then prove if any falls, the next one does too.
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**Combinations:** Ways to choose items from a set where ORDER *doesn't* matter.
Ex: Picking 2 from {A,B,C} is {A,B}, {A,C}, {B,C} (3 ways). Formula: C(n,k)=n!/(k!(n-k)!)
Pro-Tip: Think groups or subsets – the sequence you pick them in doesn't change the group!
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Alright, future engineers!
**Set Union (A U B):** All elements in set A, set B, or both.
Ex: If A={1,2}, B={2,3}, then A U B = {1,2,3}.
Pro-Tip: Think OR! An element is in A U B if it's in A OR B.
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Alright, future engineers!
**Modular Arithmetic (a mod n):** Finds the remainder when 'a' is divided by 'n'. Numbers wrap around.
Ex: 7 mod 3 = 1 (because 7 = 2*3 + 1).
Pro-Tip: Crucial for time math (clocks!), cryptography, and hashing algorithms.
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Alright, future engineers!
**Implication (P -> Q):** If P, then Q. It's only FALSE if P is True & Q is False.
Ex: 'If it rains (P), then ground is wet (Q).'
Pro-Tip: If P is false, P->Q is ALWAYS true! Crucial for understanding logic.
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Alright, future engineers!
**Combination:** A selection of items where the order of selection *doesn't matter*.
Ex: Choosing 3 people from 10 for a committee: C(10,3) = 120.
Pro-Tip: Think 'C' for 'Committee' – selecting a group where roles aren't specified.
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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