Fractionsuch

Fractionsuch is a mathematical notation used to denote the ratio of two quantities evaluated over a subset of a domain defined by a predicate. Formally, given a domain D, two functions A and B from D to real numbers, and a predicate P on D, the expression fractionsuch of A and B with respect to P is the value of (sum of A(x) over all x in D with P(x)) divided by (sum of B(x) over all x in D with P(x)), provided the denominator sum is nonzero. If the denominator sum is zero, the expression is undefined or assigned a conventional value by agreement.

Notation and usage commonly write as A fractionsuch_P B. The predicate P is read as “such that P.” This construction generalizes simple fractions by restricting the counting to elements that satisfy P; it is analogous to a conditional ratio or a weighted average, but without normalizing by the total mass outside P.

Properties and examples: Basic properties include linearity in the numerator and denominator under suitable conditions; if P is always true, fractionsuch reduces to the ordinary ratio A/B. If P isolates a subset with equal total sums in numerator and denominator, the value is preserved. Example: in a dataset of items, A(x) and B(x) could be scores; with P selecting items of interest, fractionsuch yields the ratio of total A to total B within that subset.

Relation and applications: Fractionsuch connects to conditional probability and weighted averages and can be used in data analysis, economics, and theoretical work to express restricted ratios succinctly. See also conditional expectation, weighted average, and ratio.