Discrete Mathematics course

1. 1.Logic and Sets

   Logic studies statements that are either true or false and how connectives combine them: and ($\land$), or ($\lor$), not ($\neg$) and if-then ($\to$). Truth tables list every case. An implication $p \to q$ is false only when $p$ is true and $q$ is false, and it is logically equivalent to its contrapositive $\neg q \to \neg p$, but not to its converse $q \to p$.

   • Contrapositive is equivalent; converse and inverse are not.
   • De Morgan: negating flips and/or (or $\cap$/$\cup$) and negates each part.
   • Inclusion-exclusion: $|A \cup B| = |A| + |B| - |A \cap B|$.
   Quick check: How many subsets does a 5-element set have?

   $32$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study logic and sets

2. 2.Mathematical Induction

   Mathematical induction proves that a statement $P(n)$ holds for every whole number $n \ge n_0$. It works like dominoes: show the first case is true (the base case), then show that whenever one case is true, the next one must be too (the inductive step). Together these push the truth through every integer.

   • State $P(n)$ clearly before starting.
   • Base case: verify $P(n_0)$ directly.
   • Inductive step: assume $P(k)$, then prove $P(k + 1)$ using it.
   Quick check: Use the formula to compute $1 + 2 + \cdots + 100$.

   $5050$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study mathematical induction

These topics belong to the course outline. Full explainers for them have not been written yet, so they are summarised here rather than given pages of their own.

Number Theory

Divisibility, primes, GCD and modular arithmetic.

Graph Theory

Vertices, edges, degree, paths and trees.

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