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Tends-to relation
The notation x→a says that the state described by a variable or index approaches the target a.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Mathematical induction
Mathematical induction proves an entire natural-number-indexed family of propositions through a base case and an induction step.
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Basic counting principles
The addition and multiplication principles combine finite counts across disjoint branches or successive choices.
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Factorial
The factorial of a natural number n is defined by 0!=1 and a successor recurrence, and equals the product of the positive integers from 1 through n.
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Absolute value
The absolute value of a real number x is its nonnegative distance from 0, taking x or -x according to the sign of x.
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Fraction
A fraction p/q is a representation of a rational number whose numerator and denominator are integers and whose denominator is nonzero.
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Ratio and proportion
This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios.
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Domain, codomain, and range
For a total function f:A→B, A is the domain, B is the codomain, and f[A] is the set of outputs actually attained.
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Function composition
The composition g∘f first evaluates f and then supplies its output to g; the intermediate value must be an allowed input of g.
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Sequence
This entry uses real sequences a:N→R indexed from 0. Sequence terms are ordered by their indices and cannot be rearranged into an unordered set.
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Permutations and combinations
When k distinct objects are selected without replacement from n distinct objects, an ordered selection is a permutation and an unordered selection is a combination.
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Coordinate plane
After fixing an origin, perpendicular oriented real axes, and units, a point in the plane is represented by an ordered pair of real numbers (x,y).
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Natural-number powers and principal square roots
This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct.
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Univariate polynomials over the rationals
A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree.
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Polynomial factorization over the rationals
Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values.
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Quadratic equations over the reals
For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct.
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Affine functions and degree-one functions
A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant.
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Graphs of real-valued functions
In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output.
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Probability on finite sample spaces
Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes.
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Quantifiers and explicit domains
Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false.
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Counterexamples and domain eligibility
A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement.
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Binary relations and set projections
A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output.
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Finite simple undirected graphs
This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors.
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Plane vectors in a fixed basis
This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise.