Discrete Mathematics & Math MCQ

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A set is an unordered collection of distinct objects (elements). {1,2,3} = {3,1,2}. Sets are defined by membership; the empty set ∅ has no elements.

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The power set P(S) is the set of all subsets of S including ∅ and S itself. For S={1,2}: P(S) = {∅, {1}, {2}, {1,2}}. |P(S)| = 2^|S| = 4.

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A proposition (statement) has a definite truth value (T or F). "2+2=4" is a proposition (T). "What time is it?" is not (it's a question). Compound propositions use connectives (∧, ∨, ¬, →, ↔).

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p ∧ q (p AND q) is true only when both p and q are true. If either is false, the conjunction is false. Truth table: T∧T=T, T∧F=F, F∧T=F, F∧F=F.

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De Morgan's Laws: NOT(A AND B) = (NOT A) OR (NOT B); NOT(A OR B) = (NOT A) AND (NOT B). Fundamental for circuit design and logical simplification.

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A function f: A→B assigns to each element in A exactly one element in B. Domain (A), codomain (B), range (actual outputs). f(x)=x² is a function; ±√x is not a function (one input, two outputs).

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Injective (1-to-1): f(a)=f(b)→a=b (no two inputs share output). Surjective (onto): range = codomain (every y has an x with f(x)=y). Bijective: both — enables invertibility.

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Strong induction: prove P(1) (base), then assume P(k) for all k≤n and prove P(n+1). Used to prove formulas like 1+2+...+n = n(n+1)/2 and properties of recursively defined sequences.

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Graphs model relationships: vertices are entities, edges are connections. Simple graphs: no self-loops or multi-edges. Directed graphs (digraphs): edges have direction. Applications: social networks, routing, scheduling.

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Trees are minimally connected graphs. Any two vertices are connected by exactly one path. Rooted trees have a designated root with parent-child relationships. Spanning trees connect all vertices of a graph.

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Relations generalize functions. R ⊆ A×B: (a,b) ∈ R means a relates to b. Properties: reflexive (aRa), symmetric (aRb→bRa), transitive (aRb∧bRc→aRc), antisymmetric (aRb∧bRa→a=b).

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Equivalence relations (=, ≡ mod n, "same parity") partition a set into disjoint equivalence classes. Every element belongs to exactly one class. Modular arithmetic partitions integers into n residue classes.

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Pigeonhole: with n+1 objects in n boxes, at least one box has 2+ objects. Generalized: n objects in k boxes → some box has ≥ ⌈n/k⌉ objects. Application: among 367 people, at least two share a birthday.

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P(n,r) = n!/(n-r)! counts ordered arrangements of r items from n. Arranging 3 of 5 people in a line: P(5,3) = 60. Order matters: ABC ≠ CBA.

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C(n,r) = ₙCᵣ = "n choose r" counts subsets of size r. Choosing 3 from 5 people for a committee: C(5,3) = 10. Order doesn't matter: {A,B,C} = {C,A,B}.

Explanation

Complement Ā = U - A = {x | x ∈ U ∧ x ∉ A}. De Morgan's laws apply to sets: (A∪B)ᶜ = Aᶜ ∩ Bᶜ and (A∩B)ᶜ = Aᶜ ∪ Bᶜ.

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17 ≡ 2 (mod 5) because 17-2=15 is divisible by 5. Used in cryptography (RSA, Diffie-Hellman), hash functions, and cyclic data structures.

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Boolean algebra (George Boole) is the foundation of digital logic and set theory. Operations: +/∪ (OR), ·/∩ (AND), ¬/complement. Identities: x+0=x, x·1=x, x+x'=1, x·x'=0.

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Hamiltonian path: visit each vertex exactly once. Hamiltonian circuit: visit each vertex exactly once and return to start. NP-complete to determine if one exists. Related to Traveling Salesman Problem.

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Euler path theorem (Königsberg bridge problem): exists iff ≤2 odd-degree vertices. If exactly 2 odd vertices: path starts at one and ends at other. If 0 odd vertices: Euler circuit (starts and ends same place).

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Graph coloring models scheduling, register allocation, and map coloring. The 4-Color Theorem: any planar graph can be colored with 4 colors. Bipartite graphs: 2-colorable. NP-hard in general.

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GCD(12,8) = 4. Computed by the Euclidean algorithm: GCD(a,b) = GCD(b, a mod b). Fundamental for fractions, modular inverse, and Euclidean algorithm-based cryptography.

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LCM(4,6) = 12. LCM(a,b) = a*b / GCD(a,b). Used in fractions (finding common denominators) and cyclic event scheduling.

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A prime p > 1 has exactly two positive divisors: 1 and p. Composite: has more divisors. 2 is the only even prime. Infinitely many primes (Euclid). Fundamental Theorem of Arithmetic: unique prime factorization.

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Binary (base 2) uses digits 0 and 1. 1011₂ = 8+0+2+1 = 11₁₀. Used in computing: all data is ultimately binary. Hexadecimal (base 16) is a shorthand for binary (4 bits per hex digit).

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Valid argument: the conclusion follows necessarily from premises (the form is a tautology). Sound argument: valid + all premises are true. Modus ponens: (p→q, p) ⊢ q is a valid argument form.

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Tautologies: p ∨ ¬p (law of excluded middle), p → p. Contradictions (always false): p ∧ ¬p. Contingencies: true for some assignments. Tautologies are the theorems of propositional logic.

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Recursive definitions: factorial(0)=1 (base), factorial(n)=n*factorial(n-1) (recursive). Fibonacci: F(1)=F(2)=1, F(n)=F(n-1)+F(n-2). Induction proves properties of recursively-defined objects.

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Lattices generalize set operations: {∩, ∪} forms a lattice on power sets. Distributive lattices (like sets) satisfy additional rules. Used in formal verification, type systems, and cryptography (lattice-based crypto).

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|A∪B∪C| = |A|+|B|+|C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|. Avoids double-counting. Applications: counting derangements, the sieve of Eratosthenes, scheduling conflicts.

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Bijections (1-1 correspondences) prove two sets have the same cardinality. Inverses always exist for bijections. Bijections from a set to itself are permutations. Cantor used bijections to compare infinite set sizes.

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Euler's formula for connected planar graphs: V - E + F = 2. For K₄ (4 vertices, 6 edges, 4 faces): 4-6+4=2. Used to prove K₅ and K₃,₃ are non-planar.

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Sequences: a₁, a₂, a₃, ... A function from ℕ to some set. Arithmetic (a_n = a + (n-1)d), geometric (a_n = ar^(n-1)), Fibonacci, etc. Convergence: whether the sequence approaches a limit.

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Generating functions: G(x) = ∑ a_n x^n. Ordinary GF for Fibonacci: F(x) = x/(1-x-x²). Exponential GF: ∑ a_n x^n/n!. Operations on GFs correspond to operations on sequences (convolution, shifting).

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⌊3.7⌋ = 3, ⌈3.2⌉ = 4, ⌊-1.3⌋ = -2, ⌈-1.3⌉ = -1. Used in algorithm analysis (e.g., ⌊log₂ n⌋ + 1 = bits to represent n), hashing (⌊n/m⌋ for bucketing), and pagination.

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Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B). Updates prior P(A) with likelihood P(B|A) given evidence B. Foundation of Bayesian statistics, Naive Bayes classifiers, and spam filtering.

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Big O: ignores constants and lower-order terms. O(n²) is worse than O(n log n) for large n. 100n is O(n), not O(100n). Used to analyze algorithm scalability independent of hardware.

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A∪B (union: in A or B), A∩B (intersection: in both), A-B or A\B (difference: in A but not B), Ā (complement: not in A). These satisfy De Morgan's laws and distributive properties.

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Bipartite graphs (2-colorable): no odd-length cycles. Used for matching problems (job assignments, marriage problem). König's theorem: in bipartite graphs, max matching = min vertex cover.

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T(n) = 2T(n/2) + n (merge sort). Solved by Master Theorem: T(n) = O(n log n). Fibonacci: F(n) = F(n-1) + F(n-2) with closed form F(n) = φⁿ/√5 (Binet's formula).

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CRT: x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7) has unique solution mod 105. Used in RSA (reduces computation via modular arithmetic), and efficient parallel computation.

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Fermat's Little Theorem: if p is prime and gcd(a,p)=1 then a^(p-1) ≡ 1 (mod p), equivalently aᵖ ≡ a (mod p). Foundation of RSA (a^φ(n) ≡ 1 mod n for Euler's theorem generalization).

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φ(n) counts integers ≤ n coprime to n. φ(p)=p-1 for prime p. φ(pq)=(p-1)(q-1) for distinct primes. Used in RSA: d·e ≡ 1 (mod φ(n)).

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Partial orders (≤, ⊆, divisibility) may leave some pairs incomparable. Total orders (⊆ where all pairs comparable): integers, reals. Hasse diagrams visualize partial orders. Used in scheduling and type systems.

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Master Theorem: compare f(n) with n^(log_b a). If f(n) = O(n^(log_b a - ε)): T(n) = Θ(n^(log_b a)). If equal: Θ(n^(log_b a) log n). If f(n) dominates: Θ(f(n)). Merge sort: T(n)=2T(n/2)+n → O(n log n).

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Both are equivalent in proving power. Strong induction: the inductive hypothesis assumes all cases up to k (not just k). Useful for recursion where f(n) depends on f(k) for multiple k<n (Fibonacci).

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A derangement is a permutation where no element stays in its original position. D(n) = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!). D(3)=2, D(4)=9. D(n)/n! → 1/e ≈ 0.368.

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S(n,k) = k*S(n-1,k) + S(n-1,k-1). S(4,2) = 7 (ways to partition {1,2,3,4} into 2 groups). Related to Bell numbers (sum over k), which count total partitions.

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Cayley's formula: T(n) = nⁿ⁻². Proved by Prüfer sequences (bijection between labeled trees and sequences of length n-2 from {1,...,n}). K₄ has 4² = 16 labeled spanning trees.

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The Laplacian L has degree matrix D minus adjacency matrix A. Any cofactor of L (determinant of (n-1)×(n-1) submatrix) counts spanning trees. Generalizes Cayley's formula for non-complete graphs.

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Formal language L over alphabet Σ: L ⊆ Σ*. Operations: concatenation, union, Kleene star. Recognized by automata (FA for regular, PDA for CFL, TM for RE). Bridges discrete math and computability theory.

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Well-ordering principle: every non-empty set of natural numbers has a least element. This is equivalent to (strong) mathematical induction. Both are axioms of ℕ, each derivable from the other.

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P(G,k) counts proper colorings using k colors. For K_n: k(k-1)(k-2)...(k-n+1). P(G,χ(G))>0 where χ(G) is chromatic number. Deletion-contraction: P(G,k)=P(G-e,k)-P(G/e,k).

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R(3,3)=6: any 2-coloring of K₆ edges has a monochromatic triangle. R(m,n) grows rapidly and is notoriously hard to compute. Erdős: "If aliens demand R(5,5) or they destroy Earth, we should compute; for R(6,6), we should attack."

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Cantor: ℝ is uncountable (diagonal argument). ℕ, ℤ, ℚ are countably infinite (ℵ₀). |ℝ| = 2^ℵ₀ = ℵ₁ (assuming CH). Between ℵ₀ and 2^ℵ₀ there is no set (Continuum Hypothesis — independent of ZFC).

Explanation

F(x) = Σ F_n x^n = x/(1-x-x²). Partial fraction decomposition gives Binet: F_n = (φⁿ - ψⁿ)/√5 where φ=(1+√5)/2, ψ=(1-√5)/2. GFs provide closed forms for many combinatorial sequences.

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Zorn's Lemma is equivalent to the Axiom of Choice (and Well-Ordering Theorem). Used to prove: every vector space has a basis, every ring has a maximal ideal, Tychonoff's theorem. Controversial but indispensable.

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Each edge contributes 2 to the degree sum (one to each endpoint). Corollary: the number of odd-degree vertices is always even. Fundamental for proving impossibility results (e.g., Königsberg bridges).

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4CT (Appel & Haken 1976): any planar map can be 4-colored. Also proved by Robertson, Sanders, Seymour, Thomas (1996). Notorious for requiring computer case-checking of 1,476 configurations.

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FTA: n = p₁^a₁ * p₂^a₂ * ... * pₖ^aₖ uniquely. Existence (proved by strong induction) and uniqueness (proved using Euclid's lemma: if p|ab and p is prime, then p|a or p|b). Foundation of number theory.

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DNF (disjunctive normal form): OR of AND-clauses (minterms). CNF (conjunctive): AND of OR-clauses (maxterms). Any Boolean function has unique canonical DNF and CNF. Used in SAT solvers and logic synthesis.

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Stirling's approximation (more precisely option d): n! ≈ √(2πn) * (n/e)^n. Option a is also correct notation. Used in algorithm analysis for log(n!) = Θ(n log n) and combinatorics. Proves information-theoretic Ω(n log n) sorting lower bound.

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Dilworth (1950): min chain cover = max antichain. Mirsky's dual: min antichain cover = max chain length. Used in scheduling (minimum processors = maximum concurrent tasks) and combinatorics.

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LEVEL has 5 letters with L repeated twice and E repeated twice, so the count of distinct arrangements is 5!/(2!·2!) = 120/4 = 30. This is the standard formula for permutations of a multiset with repeated elements.

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The classic contradiction proof assumes √2 is rational in lowest terms a/b, shows 2b² = a² forces a even, then b even too, contradicting gcd(a,b)=1. This pattern (assume the negation, derive a contradiction) is the template for irrationality proofs.

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A connected graph on n vertices needs at least n-1 edges, achieved exactly by a tree. For n=7, the minimum is 6 edges; fewer would disconnect the graph (it would form a forest with multiple components).

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Antisymmetry forbids two distinct elements from relating to each other in both directions: aRb and bRa together force a = b. The divisibility relation "a | b" on positive integers is antisymmetric, which is part of why it forms a partial order.

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Total committees: C(10,4) = 210. Committees containing both specific people together: C(8,2) = 28 (choosing the remaining 2 members from the other 8 people). Valid committees = 210 - 28 = 182, an application of complementary counting.

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Odd cycles cannot be 2-colored because the last edge would force two adjacent vertices to share a color, but they can always be properly colored with 3 colors. So χ(C₅) = 3, while even cycles have χ = 2 (they are bipartite).

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252 = 2(105) + 42; 105 = 2(42) + 21; 42 = 2(21) + 0. The last nonzero remainder is 21, so gcd(252,105) = 21. Each step replaces (a,b) with (b, a mod b) until the remainder is zero.

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For a_n = 5a_{n-1} - 6a_{n-2}, the characteristic equation x² - 5x + 6 = 0 factors as (x-2)(x-3) = 0, giving roots 2 and 3. The general solution is a_n = A·2ⁿ + B·3ⁿ, with A and B determined by initial conditions.

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The contrapositive of p → q is ¬q → ¬p, which is logically equivalent to the original statement. "If the ground is not wet, then it does not rain" preserves the same truth value as "If it rains, the ground is wet," unlike the converse (option a) or inverse (option c).

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Total functions from a 4-set to a 2-set: 2⁴ = 16. Subtract the 2 constant functions (mapping everything to one element), which are not onto: 16 - 2 = 14. This matches the inclusion-exclusion formula for surjections: Σ(-1)^k C(2,k)(2-k)⁴.

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By the handshaking lemma, the sum of all vertex degrees must be even (it equals twice the number of edges). The sequence (3,3,2,1) sums to 9, an odd number, so no simple graph can have it; the other sequences all have even sums and are realizable.

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By the Binomial Theorem, setting x=y=1 in (x+y)ⁿ = Σ C(n,k) xᵏ y^(n-k) gives Σ C(n,k) = 2ⁿ. This counts all subsets of an n-element set, since C(n,k) counts the k-element subsets and there are 2ⁿ subsets total.

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Proof by minimal counterexample assumes the set of counterexamples is non-empty, invokes the well-ordering principle to pick the smallest one, and derives a contradiction by finding an even smaller counterexample. It is logically equivalent to strong induction.

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Each of the 5 distinct books independently chooses one of 3 shelves, giving 3^5 = 243 distributions. This is the standard counting rule for placing distinguishable objects into distinguishable bins with no restriction.

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Kruskal's algorithm sorts all edges by weight and greedily adds each edge to the growing forest as long as it does not form a cycle (checked via union-find), producing a minimum spanning tree. Prim's algorithm achieves the same goal by growing a single tree from a vertex.

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We need x such that 3x ≡ 1 (mod 7). Testing: 3·5 = 15 = 14+1 ≡ 1 (mod 7), so x = 5. The inverse exists because gcd(3,7) = 1; it can also be found via the Extended Euclidean Algorithm.

Explanation

Negating ∀x∃y P(x,y) flips each quantifier and negates the predicate, giving ∃x∀y ¬P(x,y): "There exists an x such that for all y, x + y ≠ 0." This quantifier-negation rule (∀↔∃ swap with inner negation) is essential for writing correct proofs and counterexamples.

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AC (Cohen 1963): independent of ZF — cannot be proved or disproved from other set theory axioms. Implies Banach-Tarski paradox (sphere decomposition). Most mathematicians accept it. Constructivists reject it.

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First Incompleteness: any consistent, complete, recursively enumerable formal system (containing PA) is incomplete — there are true but unprovable statements. Second: such a system cannot prove its own consistency. Profound limits on formal mathematics.

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Ramsey theory: "complete disorder is impossible." Van der Waerden: k-color a large enough set of integers → monochromatic arithmetic progression. Ramsey: large K_n → monochromatic K_m. Hales-Jewett: n-d tic-tac-toe always has a winning line.

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LLL (symmetric version): if each bad event has probability ≤ p and depends on ≤ d others, and ep(d+1) ≤ 1, then with positive probability no bad event occurs. Non-constructive; algorithmic version (Moser-Tardos) gives efficient algorithm.

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Erdős's probabilistic method: if Pr(good structure) > 0, then good structure exists. Non-constructive but powerful. Examples: Ramsey lower bounds, existence of high-girth high-chromatic graphs, graph discrepancy results.

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Spectral graph theory: λ₁ (largest Laplacian eigenvalue) bounds chromatic number (χ ≤ λ₁+1). λ₂ (Fiedler value) measures connectivity/mixing. Expanders have large spectral gap. Used in clustering (spectral clustering, NCut).

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T(G;x,y) = Σ (x-1)^(r-r(A)) (y-1)^(|A|-r(A)). Specializations: T(G;1+λ,1)=P(G,1+λ) (chromatic), T(G;1,2)=spanning trees, T(G;2,1)=acyclic orientations. #P-hard to compute in general.

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E[comparisons] = Σᵢ<ⱼ 2/(j-i+1) = 2n(H_n - 1) ≈ 2n ln n. Elements i,j compare iff one of them is the first pivot chosen from {i,...,j}. This linearity-of-expectation proof is elegant and standard.

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Fagin (1974): NP = ∃SO on ordered finite structures. Immerman: P = FO+LFP (first-order logic + least fixpoint). This gives machine-independent characterizations: complexity classes are exactly certain logics.

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RSA: n=pq, φ(n)=(p-1)(q-1), e·d ≡ 1 (mod φ(n)), encrypt: c = m^e mod n, decrypt: m = c^d mod n = m^(ed) mod n = m^(1+kφ(n)) ≡ m (Euler's theorem). Security: factoring n to find φ(n) is believed hard.

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Inclusion-exclusion alternates: P(∪) ≤ Σ P(Aᵢ) (union bound), P(∪) ≥ Σ P(Aᵢ) - Σ P(Aᵢ∩Aⱼ), etc. Odd sums upper bound, even sums lower bound. Used in derangements, hat-check problems, and prime sieve analysis.

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Spanning trees ↔ effective resistance ↔ random walk hitting times. The effective resistance R(u,v) = sum of 1/eigenvalues weighted by eigenvector differences. These connections power spectral sparsification algorithms.

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Matroids (Whitney 1935): ground set + independent sets satisfying hereditary and augmentation properties. The greedy algorithm on a matroid always finds a maximum weight basis. Graphic (spanning trees), linear, uniform, partition, and transversal matroids.

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Category theory: objects + morphisms. Functor F:C→D maps objects to objects and morphisms to morphisms preserving composition. Natural transformation η:F⟹G maps components faithfully. Monads = functors with unit and multiplication.

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ϑ(G) (Lovász 1979): computable by SDP in polynomial time. ϑ(C₅) = √5 = Shannon capacity of the pentagonal channel. Separates graph parameters that are NP-hard to compute. Pioneered connection between combinatorics and semidefinite programming.

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ANF over GF(2): each Boolean function has unique multilinear XOR-of-products representation. The degree of the ANF measures non-linearity. Low-degree ANF means susceptibility to algebraic attacks in cryptography (correlation attacks on stream ciphers).

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Möbius function μ on a poset: μ(x,x)=1, μ(x,y)=-Σ μ(x,z) for x≤z<y. Möbius inversion: f(x)=Σg(y) ↔ g(x)=Σ f(y)μ(y,x). Generalizes PIE; specializes to number-theoretic μ on divisibility poset.

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Szemerédi regularity lemma (1975): any dense graph has an ε-regular partition. Applications: Szemerédi's theorem (arithmetic progressions in dense sets), graph removal lemma, property testing. Tower-type bounds but non-constructive.

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Diffie-Hellman relies on a one-way function: computing gˣ mod p is fast (repeated squaring), but recovering x from gˣ mod p — the discrete logarithm — has no known efficient classical algorithm for well-chosen groups. Both parties compute g^(ab) mod p from exchanged values g^a and g^b without ever transmitting a or b.

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The associated homogeneous recurrence a_n = 2a_{n-1} has characteristic root 2, with general solution A·2ⁿ. Since the forcing term 2ⁿ shares this base ("resonance"), substituting the naive trial A·2ⁿ makes both sides cancel identically, giving no information. Multiplying by n (trial A·n·2ⁿ) breaks the degeneracy and yields a valid particular solution — the standard resonance rule for linear non-homogeneous recurrences and differential equations alike.