Draw a Venn diagram and decide region by region what belongs to the set expression.
For counting problems, fill in the regions from the centre outwards: all sets, then pairs, then single sets, then the outside.
Use inclusion–exclusion as a check: |A ∪ B| = |A| + |B| − |A ∩ B|.
For a relation, test each property on every pair and give one counterexample when it fails.
For a function, count the arrows arriving at each element of B: at most one means injective, at least one surjective.
Graph theory
Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij
Method
Write down the degree of every vertex; the degrees add up to twice the number of edges.
For Euler questions, check that the graph is connected and count the vertices with odd degree.
For trees, use n − 1 edges and the handshake lemma.
For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).
Number theory & modular arithmetic
Division with remainder
a = q·n + r, 0 ≤ r < n
Euclid
gcd(a, b) = gcd(b, a mod b)
Least common multiple
lcm(a, b) = a·b/gcd(a, b)
Inverse
a·a⁻¹ ≡ 1 (mod n), gcd(a, n) = 1
Linear congruence
a·x ≡ b → x ≡ a⁻¹·b (mod n)
Euler’s totient
φ(n) = n·∏(1 − 1/p), φ(pᵏ) = pᵏ − pᵏ⁻¹
Fermat
aᵖ⁻¹ ≡ 1 (mod p)
RSA
n = pq, φ = (p − 1)(q − 1), d = e⁻¹ mod φ, c = mᵉ mod n, m = cᵈ mod n
Method
Reduce every number modulo n first; remainders lie between 0 and n − 1.
For a gcd, divide repeatedly with Euclid’s algorithm; the last nonzero remainder is the gcd.
For an inverse, run the extended algorithm and keep the coefficient of a; add n if it is negative.
Solve a·x ≡ b by multiplying with a⁻¹; combine two congruences by writing x = r₁ + m₁k.
For powers, write the exponent in binary and square repeatedly; in RSA, d = e⁻¹ mod φ(n) with φ(n) = (p − 1)(q − 1).
2Practice problems
1.Propositional logic & proofsMedium
Is (q → p) ↔ (p → q) a tautology, a contradiction or a contingency? Click your answer.
(q → p) ↔ (p → q)
1tautology
2contradiction
3contingency
2.Propositional logic & proofsHard
The argument with premises b → a; c → a and conclusion b → c is invalid. Click the row of the truth table that is a counterexample.
b → ac → a∴ b → c
1a = 1, b = 1, c = 1
2a = 1, b = 1, c = 0
3a = 1, b = 0, c = 1
4a = 1, b = 0, c = 0
5a = 0, b = 1, c = 1
6a = 0, b = 1, c = 0
7a = 0, b = 0, c = 1
8a = 0, b = 0, c = 0
3.Sets, relations & functionsMedium
R = {(1, 2), (2, 2), (3, 1), (3, 3), (4, 2)} is a relation on {1, 2, 3, 4}. What is the smallest number k of ordered pairs that must be added to R to make it transitive (the size of the transitive closure minus |R|)?
R = {(1, 2), (2, 2), (3, 1), (3, 3), (4, 2)}
k =
4.Sets, relations & functionsHard
An equivalence relation R on {1, …, 7} has the classes {1, 4, 7}, {6}, {2, 3}, {5}. How many ordered pairs does R contain? Call the number |R|.
{1, 4, 7}{6}{2, 3}{5}
|R| =
5.Graph theoryMedium
Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.
|V| = 6|E| = 7
1Euler circuit
2Euler path, no circuit
3neither
6.Graph theoryHard
Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.
start: Aend: F
d(F) =
7.Number theory & modular arithmeticMedium
Find the inverse of 10 modulo 29: the number x with 0 < x < 29 and 10·x ≡ 1 (mod 29).
a = 10n = 29
10−1 mod 29 =
8.Number theory & modular arithmeticHard
An RSA key uses p = 41, q = 43 and e = 11. Decrypt the ciphertext c = 733: find the message m.
p = 41q = 43e = 11c = 733
m =
3Solutions
1.Propositional logic & proofs
Answer: contingency
Logical equivalence: two formulas are equivalent (≡) when they have the same value in every row of the truth table. A tautology is 1 in every row, a contradiction is 0 in every row, and a contingency is neither. Useful laws: De Morgan, ¬(a ∧ b) ≡ ¬a ∨ ¬b and ¬(a ∨ b) ≡ ¬a ∧ ¬b; a → b ≡ ¬a ∨ b ≡ ¬b → ¬a (contrapositive); ¬(a → b) ≡ a ∧ ¬b; and the distributive laws, a ∨ (b ∧ c) ≡ (a ∨ b) ∧ (a ∨ c).
Truth table row by row
p = 1, q = 1: q → p = 1, p → q = 1, (q → p) ↔ (p → q) = 1
p = 1, q = 0: q → p = 1, p → q = 0, (q → p) ↔ (p → q) = 0
p = 0, q = 1: q → p = 0, p → q = 1, (q → p) ↔ (p → q) = 0
p = 0, q = 0: q → p = 1, p → q = 1, (q → p) ↔ (p → q) = 1
Classify
(q → p) ↔ (p → q): 1, 0, 0, 1
1 in 2 of the 4 rows, 0 in the others: a contingency.
Answer
contingency
2.Propositional logic & proofs
Answer: a = 1, b = 1, c = 0
Quantifiers and arguments: ¬∀x P(x) ≡ ∃x ¬P(x) and ¬∃x P(x) ≡ ∀x ¬P(x). Move a negation inwards, swap every quantifier it passes, and negate the inner formula with the propositional laws, such as ¬(a → b) ≡ a ∧ ¬b. An argument is valid when the conclusion is 1 in every row in which all premises are 1. A single row with true premises and a false conclusion is a counterexample and shows the argument is invalid.
Premises and conclusion row by row
a = 1, b = 1, c = 1: 1 1 | 1
a = 1, b = 1, c = 0: 1 1 | 0
a = 1, b = 0, c = 1: 1 1 | 1
a = 1, b = 0, c = 0: 1 1 | 1
a = 0, b = 1, c = 1: 0 0 | 1
a = 0, b = 1, c = 0: 0 1 | 0
a = 0, b = 0, c = 1: 1 0 | 1
a = 0, b = 0, c = 0: 1 1 | 1
Counterexample
a = 1, b = 1, c = 0
Answer
a = 1, b = 1, c = 0
An argument with this form looks like a valid rule, but one row is enough to show that it is not.
3.Sets, relations & functions
Answer: k = 1
Relations: a relation R on a set X is a set of ordered pairs (x, y). R is reflexive when (x, x) ∈ R for every x in X; symmetric when (x, y) ∈ R always comes with (y, x) ∈ R; antisymmetric when (x, y) and (y, x) are both in R only if x = y; and transitive when (x, y) and (y, z) in R force (x, z) ∈ R. An equivalence relation is reflexive, symmetric and transitive, and splits X into disjoint classes.
Pairs to add
{(3, 2)}
Count
k = 1
Answer
k = 1
4.Sets, relations & functions
Answer: |R| = 15
Relations: a relation R on a set X is a set of ordered pairs (x, y). R is reflexive when (x, x) ∈ R for every x in X; symmetric when (x, y) ∈ R always comes with (y, x) ∈ R; antisymmetric when (x, y) and (y, x) are both in R only if x = y; and transitive when (x, y) and (y, z) in R force (x, z) ∈ R. An equivalence relation is reflexive, symmetric and transitive, and splits X into disjoint classes.
Pairs per class
|R| = 3² + 1² + 2² + 1² = 15
Answer
|R| = 15
5.Graph theory
Answer: neither
Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.
Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.
Dijkstra round by round
Each line: the vertex settled, its final distance, and the neighbours whose distance improves.
A (0): B: 11, C: 8
C (8): E: 15
B (11): D: 26, E: 14
E (14): F: 29
D (26)
F (29)
Shortest path
A → B → E → F
d(F) = 29
Answer
d(F) = 29
7.Number theory & modular arithmetic
Answer: 10−1 mod 29 = 3
Inverses and congruences: a has an inverse modulo n exactly when gcd(a, n) = 1. The extended Euclidean algorithm writes 1 = s·n + t·a, and then a⁻¹ ≡ t (mod n). A congruence a·x ≡ b (mod n) with gcd(a, n) = 1 has the solution x ≡ a⁻¹·b. Two congruences x ≡ r₁ (mod m₁) and x ≡ r₂ (mod m₂) with coprime moduli have one solution modulo m₁·m₂ (Chinese remainder theorem). Euler’s φ(n) counts the numbers from 1 to n that are coprime to n: φ(n) = n·∏(1 − 1/p).
Extended Euclidean algorithm
Each remainder r is written as r ≡ t·10 (mod 29).
r = 29, t = 0
r = 10, q = 2, t = 1
r = 9, q = 1, t = −2
r = 1, q = 9, t = 3
Inverse
1 ≡ 3·10 (mod 29)
x = 3
check: 10·3 = 30 = 1·29 + 1
Answer
10−1 mod 29 = 3
8.Number theory & modular arithmetic
Answer: m = 1071
Modular powers and RSA: compute aᵏ mod n by repeated squaring: write k in binary, square, reduce after every step and multiply the powers whose bit is 1. Fermat: aᵖ⁻¹ ≡ 1 (mod p) for a prime p not dividing a, so the exponent can be reduced modulo p − 1. RSA: choose primes p and q, n = p·q and φ(n) = (p − 1)(q − 1); the public exponent e is coprime to φ(n), and the private exponent is d = e⁻¹ mod φ(n). Encrypt with c = mᵉ mod n and decrypt with m = cᵈ mod n.