Tags: #◇Computer Science #◇CCDSTRU #◇Discrete Mathematics #◇Reviewer #◇Finals

CCDSTRU Comprehensive Finals Reviewer

◇Overview

This reviewer synthesizes Discrete Structures (CCDSTRU), establishing the rigorous foundations of mathematical logic, set theory, relations and functions, mathematical induction (weak and strong), elementary number theory (Euclidean algorithm, modular arithmetic, Chinese Remainder Theorem), and combinatorics.

◇1. Core Logic & Equivalence Laws

Law Name	Equivalence Form	Notes
Implication Law	$p \to q \equiv \neg p \lor q$	Fundamental for contrapositive and deduction proofs
Contrapositive	$p \to q \equiv \neg q \to \neg p$	Equivalent truth value; converse $q \to p$ is NOT equivalent
De Morgan's Laws	$\neg(p \land q) \equiv \neg p \lor \neg q$ $\neg(p \lor q) \equiv \neg p \land \neg q$	Swaps conjunction and disjunction under negation
Biconditional Law	$p \leftrightarrow q \equiv (p \to q) \land (q \to p)$	Proven by showing both directions hold
Distributive Laws	$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$	Boolean algebra analog to arithmetic multiplication

◇2. Relations & Properties

Given a relation $R$ on a set $A$ :

Reflexive: $\forall x \in A, (x, x) \in R$ .
Symmetric: $\forall x, y \in A, (x, y) \in R \implies (y, x) \in R$ .
Antisymmetric: $\forall x, y \in A, ((x, y) \in R \land (y, x) \in R) \implies x = y$ .
Transitive: $\forall x, y, z \in A, ((x, y) \in R \land (y, z) \in R) \implies (x, z) \in R$ .

Relational Classifications

Equivalence Relation: Reflexive, Symmetric, and Transitive. Partitions $A$ into disjoint equivalence classes $[x]$ .
Partial Order (Poset): Reflexive, Antisymmetric, and Transitive (e.g., $(\mathbb{Z}, \le)$ , $(\mathcal{P}(S), \subseteq)$ ).

◇3. Mathematical Induction Framework

Weak Induction: Assume $P(k)$ holds to prove $P(k+1)$ .
Strong Induction: Assume $P(i)$ holds for all $n_0 \le i \le k$ to prove $P(k+1)$ (essential for prime factorization, game theory, recurrence).

◇4. Number Theory & Modular Arithmetic Matrix

Concept	Definition / Formula	Algorithmic Tool
Divisibility	$a \mid b \iff \exists k \in \mathbb{Z}, b = ak$	Division Algorithm: $a = dq + r, 0 \le r < d$
GCD & Bézout's Identity	$\gcd(a, b) = sa + tb$ for $s, t \in \mathbb{Z}$	Extended Euclidean Algorithm
Modular Inverse	$ax \equiv 1 \pmod m \iff \gcd(a, m) = 1$	Solve $as + mt = 1 \implies a^{-1} \equiv s \pmod m$
Linear Congruence	$ax \equiv b \pmod m$	Has solutions $\iff \gcd(a, m) \mid b$
Chinese Remainder Theorem	$x \equiv a_i \pmod{m_i}$ where $\gcd(m_i, m_j) = 1$	Unique solution modulo $M = \prod m_i$

◇5. High-Yield Exam Checklist

Does converse ( $q \to p$ ) equal the original conditional ( $p \to q$ )? NO.
For strong induction, did you verify enough base cases for the recurrence depth?
Does a modular inverse exist if $\gcd(a, m) \ne 1$ ? NO.
Did you remember that an equivalence relation requires symmetry, whereas a partial order requires antisymmetry?
Is a relation antisymmetric if it has no pairs with $x \ne y$ ? YES (vacuously true).

Previous Topic: ◇CCPROG1
Next Topic: ◇CCDSALG
Prerequisites: ◇Discrete Mathematics, Logic, Set Theory
Cross-Curricular Connections: ◇CSMATH1, ◇CSALGCM

[REV] CCDSTRU COMPREHENSIVE FINALS REVIEWER