Logic, sets and proofs problems
148 problems · Proof techniques, quantifiers, sets, maps and cardinality.
Ex.182Congruence is an equivalence relation
Relations
L1
Ex.183De Morgan's law for sets
Sets and maps
L1
Ex.186Injective or surjective?
Sets and maps
L1
Ex.218Cardinality of finite sets
Countability and cardinality
High school
Ex.219$\mathbb{N}$ is countably infinite
Countability and cardinality
High school
Ex.248Proof by counterexample
Proof techniques
High school
Ex.249De Morgan's laws
Proof techniques
High school
Ex.250Direct proof
Proof techniques
High school
Ex.251Geometric sum by induction
Induction
High school
Ex.252Proof by induction: sum of first $n$ integers
Induction
High school
Ex.253Summary of induction
Induction
High school
Ex.267The existential quantifier
Quantifiers and negation
High school
Ex.268The universal quantifier
Quantifiers and negation
High school
Ex.269Double negation
Quantifiers and negation
High school
Ex.270Proving an existential statement
Quantifiers and negation
High school
Ex.271Universal over a finite set
Quantifiers and negation
High school
Ex.272Binary relations
Relations
High school
Ex.273Reflexive relations
Relations
High school
Ex.274Symmetric relations
Relations
High school
Ex.275Transitive relations
Relations
High school
Ex.277Basic set operations
Sets and maps
High school
Ex.278The empty set
Sets and maps
High school
Ex.279Subset relations
Sets and maps
High school
Ex.286Cartesian product with seven elements
Sets and maps
L1
Ex.323Bernoulli's inequality
Induction
L1
Ex.329Negating convergence
Quantifiers and negation
L1
Ex.337Sum of squares by induction
Induction
High school
Ex.341The integers are countable
Countability and cardinality
L1
Ex.416Countable sets
Countability and cardinality
L1
Ex.417Hilbert's hotel
Countability and cardinality
High school
Ex.418Infinite sets
Countability and cardinality
High school
Ex.419$\mathbb{Z}$ is countable
Countability and cardinality
L1
Ex.420Summary of cardinality
Countability and cardinality
High school
Ex.495Proof by contradiction
Proof techniques
High school
Ex.496Proof by contraposition
Proof techniques
High school
Ex.497Contrapositive with primes
Proof techniques
L1
Ex.498Converse, inverse, contrapositive
Proof techniques
High school
Ex.499Bernoulli's inequality by induction
Induction
L1
Ex.500Sum of squares by induction
Induction
High school
Ex.501Proof by cases
Proof techniques
High school
Ex.502A parity case analysis
Proof techniques
High school
Ex.503A tautology
Proof techniques
High school
Ex.504Negating quantifiers
Proof techniques
High school
Ex.505Summary of logic and proof techniques
Proof techniques
High school
Ex.558Bounded quantifiers
Quantifiers and negation
L1
Ex.559Quantifier over a finite set
Quantifiers and negation
High school
Ex.560Negating an existential statement
Quantifiers and negation
High school
Ex.561Negating a universal statement
Quantifiers and negation
High school
Ex.562Summary of quantifiers and negation
Quantifiers and negation
High school
Ex.563The uniqueness quantifier
Quantifiers and negation
L1
Ex.564Refuting a universal statement
Quantifiers and negation
High school
Ex.565Antisymmetric relations
Relations
High school
Ex.566Equivalence classes
Relations
High school
Ex.567Equivalence relations
Relations
High school
Ex.568Partial orders
Relations
L1
Ex.572Cardinality of finite sets
Sets and maps
High school
Ex.573Cartesian product
Sets and maps
High school
Ex.574Complement laws
Sets and maps
High school
Ex.575Distributive laws for sets
Sets and maps
High school
Ex.576The power set
Sets and maps
High school
Ex.577Relations and their graphs
Sets and maps
L1
Ex.578Symmetric difference
Sets and maps
L1
Ex.626Inclusion and complement
Sets and maps
L1
Ex.627Inclusion and union with the complement
Sets and maps
L1
Ex.628Cardinality of the power set
Sets and maps
L1
Ex.817The cardinal $\aleph_0$
Countability and cardinality
L1
Ex.818The cardinality of the continuum
Countability and cardinality
L1
Ex.819$\mathbb{Q}$ is dense but countable
Countability and cardinality
L1
Ex.820Cantor's diagonal argument
Countability and cardinality
L1
Ex.821Comparing cardinalities
Countability and cardinality
L1
Ex.822$\mathbb{N}^2$ is countable
Countability and cardinality
L1
Ex.823Countable product
Countability and cardinality
L1
Ex.824$\mathbb{Q}$ is countable
Countability and cardinality
L1
Ex.1031Proving an equivalence
Proof techniques
High school
Ex.1032Coins and strong induction
Induction
L1
Ex.1033Full binary tree node count
Induction
L1
Ex.1034Divisibility by induction
Induction
L1
Ex.1035An identity for Fibonacci numbers
Induction
L1
Ex.1036Towers of Hanoi
Induction
L1
Ex.1037Exponential inequality by induction
Induction
L1
Ex.1038Fibonacci growth bound
Induction
L1
Ex.1039$n!$ vs $2^n$
Induction
L1
Ex.1040Induction and periodicity
Induction
L1
Ex.1041Product by induction
Induction
L1
Ex.1042Strong induction
Induction
L1
Ex.1043Sum of cubes
Induction
L1
Ex.1044A factorial identity
Induction
L1
Ex.1045Existence proofs
Proof techniques
L1
Ex.1046Order of quantifiers
Proof techniques
L1
Ex.1150Quantifier absorption
Quantifiers and negation
L1
Ex.1151Renaming bound variables
Quantifiers and negation
L1
Ex.1152De Morgan's laws for quantifiers
Quantifiers and negation
L1
Ex.1153The $\varepsilon$-$\delta$ definition of limit
Quantifiers and negation
L1
Ex.1154Existence and uniqueness proof
Quantifiers and negation
L1
Ex.1155A false $\exists\forall$ statement
Quantifiers and negation
L1
Ex.1156A quantified implication
Quantifiers and negation
L1
Ex.1157Free and bound variables
Quantifiers and negation
L1
Ex.1158Quantifiers in induction proofs
Quantifiers and negation
L1
Ex.1159Quantifiers in convergence of sequences
Quantifiers and negation
L1
Ex.1160Negating nested quantifiers
Quantifiers and negation
L1
Ex.1161Negating the uniqueness quantifier
Quantifiers and negation
L1
Ex.1162Order of quantifiers matters
Quantifiers and negation
L1
Ex.1163Vacuous truth
Quantifiers and negation
L1
Ex.1164Hasse diagram
Relations
L1
Ex.1165Equivalence classes form a partition
Relations
L1
Ex.1166Partitions induce equivalence relations
Relations
L1
Ex.1167Total orders
Relations
L1
Ex.1176Arbitrary unions and intersections
Sets and maps
L1
Ex.1177Equivalence classes and partitions
Sets and maps
L1
Ex.1178Equivalence relations
Sets and maps
L1
Ex.1179Partial orders
Sets and maps
L1
Ex.1180Total orders
Relations
L1
Ex.1283Two implications imply the negation of $p$
Proof techniques
L1
Ex.1284The barber's paradox
Quantifiers and negation
L1
Ex.1285Three numbers with signs
Quantifiers and negation
L1
Ex.1463Cantor's theorem
Countability and cardinality
L3
Ex.1558Cantor–Bernstein theorem
Countability and cardinality
L1
Ex.1559The Cantor set is uncountable
Countability and cardinality
M1
Ex.1560Cantor's theorem
Countability and cardinality
L1
Ex.1561Finite subsets of $\mathbb{N}$
Countability and cardinality
M1
Ex.1562Ordinals vs cardinals
Countability and cardinality
M1
Ex.1563$|\mathcal{P}(\mathbb{N})|=|\mathbb{R}|$
Countability and cardinality
L1
Ex.1564$\mathbb{R}$ is uncountable
Countability and cardinality
L1
Ex.1565Countable union of countable sets
Countability and cardinality
L1
Ex.1755Euler characteristic by induction
Induction
L1
Ex.1756Determinant of a matrix by induction
Induction
L1
Ex.1757Existence of quotient and remainder
Induction
L1
Ex.1758Every positive integer is a sum of non-consecutive Fibonacci numbers
Induction
Olympiad
Ex.1759Representation of integers
Induction
Olympiad
Ex.1760Tiling by induction
Induction
Olympiad
Ex.1761Russell's paradox (statement)
Proof techniques
L1
Ex.1857Alternation depth of quantifiers
Quantifiers and negation
L1
Ex.1858Scope of quantifiers
Quantifiers and negation
L1
Ex.1859When distribution fails
Quantifiers and negation
L1
Ex.1860Proving a quantified statement with $\exists\forall$
Quantifiers and negation
Olympiad
Ex.1861Logical equivalence of quantified statements
Quantifiers and negation
L1
Ex.1862Negating a complex statement
Quantifiers and negation
L1
Ex.1863Prenex normal form
Quantifiers and negation
L1
Ex.1864A statement with a specific structure
Quantifiers and negation
L1
Ex.1882Russell's paradox
Sets and maps
L1
Ex.2096Baire category theorem (statement)
Countability and cardinality
M1
Ex.2097Cantor function
Countability and cardinality
M1
Ex.2098Trichotomy of cardinals (statement)
Countability and cardinality
M1
Ex.2099The continuum hypothesis (statement)
Countability and cardinality
M1
Ex.2100The generalized continuum hypothesis (statement)
Countability and cardinality
M1
Ex.2101Infinite subsets of $\mathbb{N}$
Countability and cardinality
M1
Ex.2102Proof of Cantor–Bernstein–Schröder
Countability and cardinality
M1
Ex.2177Real induction (statement)
Induction
M1
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