Subset

A subset is a set whose elements are all contained within another set. Expressed as a relation, a set \(A\) is a subset of a set \(B\) if all elements of \(A\) are also elements of \(B\), written as \(A \subseteq B.\) If \(A\) has an element that \(B\) does not have, then \(A \nsubseteq B.\)

For every set \(S\), if the universal set is \(U\), then \(\emptyset \subseteq S \subseteq U.\)

You can prove that two sets are equal if you show that each is a subset of the other. This method is called "double-inclusion."

Contents

Proper subsets

For two sets \(A\) and \(B\), if \(A \subseteq B\), and there is an element which \(B\) has that \(A\) does not have, then \(A\) is considered a proper subset of \(B\), expressed as \(A \subset B.\) A proper subset of a set cannot be equal to that set. Here, the superset is larger; it has a greater cardinality.

⚠ Notice that the symbols for subset and proper subset, \(\subseteq\) and \(\subset\), are analogous to the inequality symbols \(\geq\) and \(\gt\), or \(\leq\) and \(\lt.\) Therefore, if \(A \subseteq B\), then \(A\) may equal \(B\), but if \(A \subset B\), \(A\) is definitely not equal to \(B.\)

Logic & Proofs

Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form

Set Theory

Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof

Functions

Function • Floor function • Ceiling function • Inverse function

Algorithms

Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm

Relations

Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class

Number Theory

Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm

Induction

Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition

Combinatorics

Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle

Graph Theory

Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm

Recursion

Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort