# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=1#results)[1](/en/knowledge/?page=1#results) 2 [3](/en/knowledge/?page=3#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=3#results) 302 results · Page 2 of 13 Knowledge object ## [Tends-to relation](/en/mko/mko-tends-to-relation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) The notation x→a says that the state described by a variable or index approaches the target a. mko-tends-to-relation · Analysis Knowledge object ## [Mathematical induction](/en/mko/mko-mathematical-induction/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) Mathematical induction proves an entire natural-number-indexed family of propositions through a base case and an induction step. mko-mathematical-induction · History, philosophy and methodology Knowledge object ## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) The addition and multiplication principles combine finite counts across disjoint branches or successive choices. mko-counting-principle · Combinatorics and discrete mathematics Knowledge object ## [Factorial](/en/mko/mko-factorial/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-factorial · Combinatorics and discrete mathematics Knowledge object ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) The absolute value of a real number x is its nonnegative distance from 0, taking x or -x according to the sign of x. mko-absolute-value · Analysis Knowledge object ## [Fraction](/en/mko/mko-fraction/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) A fraction p/q is a representation of a rational number whose numerator and denominator are integers and whose denominator is nonzero. mko-fraction · Arithmetic and number systems Knowledge object ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios. mko-ratio-proportion · Arithmetic and number systems Knowledge object ## [Domain, codomain, and range](/en/mko/mko-domain-codomain-range/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-domain-codomain-range · Foundations and logic Knowledge object ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) The composition g∘f first evaluates f and then supplies its output to g; the intermediate value must be an allowed input of g. mko-composition · Foundations and logic Knowledge object ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-sequence · Analysis Knowledge object ## [Permutations and combinations](/en/mko/mko-permutation-combination/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-permutation-combination · Combinatorics and discrete mathematics Knowledge object ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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). mko-coordinate-plane · Geometry Knowledge object ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-power-root · Algebra Knowledge object ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree. mko-polynomial · Algebra Knowledge object ## [Polynomial factorization over the rationals](/en/mko/mko-factorization/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values. mko-factorization · Algebra Knowledge object ## [Quadratic equations over the reals](/en/mko/mko-quadratic-equation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-quadratic-equation · Algebra Knowledge object ## [Affine functions and degree-one functions](/en/mko/mko-linear-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant. mko-linear-function · Algebra Knowledge object ## [Graphs of real-valued functions](/en/mko/mko-function-graph/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-function-graph · Geometry Knowledge object ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes. mko-probability · Probability and statistics Knowledge object ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-quantifier · Foundations and logic Knowledge object ## [Counterexamples and domain eligibility](/en/mko/mko-counterexample/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-counterexample · History, philosophy and methodology Knowledge object ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) 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. mko-relation · Foundations and logic Knowledge object ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors. mko-graph · Combinatorics and discrete mathematics Knowledge object ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fknowledge%2F%3Fpage%3D2) This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise. mko-vector · Algebra [← Previous](/en/knowledge/?page=1#results)[1](/en/knowledge/?page=1#results) 2 [3](/en/knowledge/?page=3#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=3#results) 302 results · Page 2 of 13