Discrete Mathematics
Key points
The Basics of Graph Theory
A graph G = (V, E) is made of a set of vertices V and a set of edges E connecting them. Edges can be directed (one-way), undirected (two-way), or carry weights such as distance or cost. The degree of a vertex is simply how many edges touch it.
Mathematical Induction
Two steps are all you need. First, show P(1) is true (the base case). Second, show that if P(k) is true then P(k+1) must be true as well (the inductive step). Together they prove P(n) for every natural number n — like knocking over the first domino and knowing each one topples the next.
Proving Σk = n(n+1)/2 by Induction
Base case, n=1: 1 = 1×2/2 = 1 ✓. Inductive step, assume Σ₁^k k = k(k+1)/2, then Σ₁^{k+1} k = k(k+1)/2 + (k+1) = (k+1)(k+2)/2 ✓ — which is exactly the formula with k+1 in place of n.
Propositional Logic and Boolean Algebra
Arithmetic on true/false values (1 and 0) using AND (∧), OR (∨), NOT (¬), and XOR (⊕). Because there are only finitely many inputs, a truth table can list every possible case. This is the foundation logic circuits and CPUs are designed on.
BFS and DFS (Searching a Graph)
Breadth-first search (BFS) fans out one layer at a time, so it's the tool for shortest paths and for "how many degrees of separation apart are these two people?" Depth-first search (DFS) follows one path as far as it goes before backing up, which suits backtracking, maze solving, and topological sorting.
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