Inequality

An inequality is a relation between two numbers that indicates their relative sizes.

If \(a > b\), then \(a\) is greater than \(b.\)
If \(a < b\), then \(a\) is less than \(b.\)
If \(a \ge b\), then \(a\) is greater than or equal to \(b\), "at least" \(b.\)
If \(a \le b\), then \(a\) is less than or equal to \(b\), "at most" \(b.\)

Now, consider their negations. If it is not true that \(a > b\), then either \(a < b\) or \(a = b\), which can be translated into \(a \le b.\) The same reasoning shows the negation of \(a < b\) is \(a \ge b.\) Similarly, if it is not true that \(a \ge b\), it is only possible for \(a < b\) to be true. Finally, if \(a \le b\) is a false statement, then \(a > b\) is a true statement.

If \(a > b\), it is also true that \(a \ge b.\) This is because the "greater than or equal to" requirement has been satisfied by "greater than" being true. You can apply the same logic to \(a < b\) to obtain \(a \le b.\)

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