Bulgaria DZI Mathematics key terms

A collection of distinct objects, called its elements, considered as a whole. Membership is written a ∈ A, and the empty set ∅ contains no elements at all.

A set A is a subset of B when every element of A also belongs to B, written A ⊆ B. If additionally B has elements outside A, the subset is called proper.

The set of all elements belonging to at least one of the given sets. For A and B it is written A ∪ B and contains everything appearing in A, in B, or in both.

The set of elements common to all given sets. For A and B it is written A ∩ B and contains exactly those objects that lie in A and in B simultaneously.

Relative to a universal set U, the complement of A consists of all elements of U that are not in A, written A′ or U \ A.

The distance of a real number from zero on the number line, always non-negative. Formally |x| equals x when x ≥ 0 and equals −x when x < 0.

A connected portion of the real line described by endpoints, such as (a, b) excluding both ends or [a, b] including both. It is the standard way to write solution sets.

A statement comparing two expressions with one of the symbols <, ≤, >, ≥. Solving it means finding all values of the variable that make the statement true.

An equation of the form ax² + bx + c = 0 with a ≠ 0. Its real solutions are found from the discriminant b² − 4ac and the quadratic formula.

For ax² + bx + c = 0 the number D = b² − 4ac. It tells how many real roots exist: two when D > 0, one when D = 0, none when D < 0.

Relations linking the roots of a quadratic to its coefficients: for ax² + bx + c = 0, the sum of roots is −b/a and the product is c/a.

An inequality in which one side is a quadratic expression, for example ax² + bx + c > 0. Its solution set is read from the sign of the parabola between and beyond its roots.

Two or more equations that must hold simultaneously for the same unknowns. A solution is an ordered set of values satisfying every equation at once.

An equation in which each variable appears to the first power only, such as ax + b = 0. It has exactly one solution when the coefficient of the variable is nonzero.

A technique for solving systems in which one equation is rearranged to express a variable, and that expression replaces the variable in the other equation.

A rule assigning to each element of a domain exactly one element of a codomain. The graph of a function passes the vertical line test.

The set of all input values for which the function is defined. Restrictions arise from division by zero, even roots of negative numbers, and logarithms of non-positive numbers.

The set of all output values the function actually attains as the input runs through its domain. It is read from the graph as the spread along the vertical axis.

The graph of a quadratic function y = ax² + bx + c, a curve opening upward when a > 0 and downward when a < 0. Its vertex gives the extreme value.

The turning point of a parabola, located at x = −b/(2a) for y = ax² + bx + c. It is the minimum point when a > 0 and the maximum when a < 0.

A function of the form f(x) = aˣ with a > 0 and a ≠ 1. It is increasing when a > 1, decreasing when 0 < a < 1, and always positive.

The exponent to which a base must be raised to obtain a given number: log_a b = c means aᶜ = b, with a > 0, a ≠ 1, b > 0.

An equation in which the unknown appears in an exponent, such as 2ˣ = 8. It is usually solved by writing both sides as powers of the same base.

An equation containing the unknown inside a logarithm, for example log₂ x = 3. Solutions must be checked against the requirement that the argument stays positive.

The natural unit of angle measure defined so that a full turn is 2π radians. One radian equals 180/π degrees, roughly 57.3 degrees.

For an acute angle in a right triangle, the ratio of the opposite leg to the hypotenuse. Extended to all angles, it is the y-coordinate on the unit circle.

For an acute angle in a right triangle, the ratio of the adjacent leg to the hypotenuse. Extended to all angles, it is the x-coordinate on the unit circle.

The ratio of sine to cosine of an angle, equal in a right triangle to the opposite leg divided by the adjacent leg. It is undefined where cosine equals zero.

A relation in any triangle: c² = a² + b² − 2ab·cos C, where C is the angle between sides a and b. It generalises the Pythagorean theorem.

In any triangle, the ratio a/sin A = b/sin B = c/sin C is constant and equals twice the circumradius. It is used when an angle and its opposite side are known.

A sequence in which each term after the first is obtained by adding a fixed difference d. The n-th term is a₁ + (n − 1)d.

A sequence in which each term after the first is obtained by multiplying by a fixed ratio q. The n-th term is a₁·qⁿ⁻¹.

The total of the first n terms of a sequence. For an arithmetic progression it is n(a₁ + aₙ)/2; for a geometric one with q ≠ 1 it is a₁(qⁿ − 1)/(q − 1).

An arrangement of objects in which order matters. The number of orderings of n distinct objects is n!, and choosing and ordering k of them gives n!/(n − k)!.

A selection of objects in which order does not matter. The number of ways to choose k objects from n is written C(n, k) and equals n!/(k!(n − k)!).

The product of all positive integers up to a given one, written n! = 1·2·…·n, with 0! defined as 1. It counts orderings of n distinct objects.

A number from 0 to 1 measuring how likely the event is. For equally likely outcomes it equals the count of favourable outcomes divided by the count of all outcomes.

The arithmetic average of a data set, found by adding all values and dividing by how many there are. It is sensitive to extreme values.

The middle value of a data set arranged in order; with an even count it is the average of the two central values. Half the data lies on each side.

In a right triangle the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². Its converse tests whether a triangle is right-angled.

Triangles with equal corresponding angles and proportional corresponding sides. The ratio of their areas equals the square of the ratio of corresponding lengths.

The amount of space enclosed by a three-dimensional body, measured in cubic units. Each solid type has its own formula, for example a prism's base area times height.

The total area of all faces of a solid, measured in square units. For a sphere of radius r it equals 4πr².

A solid with a polygonal base and triangular faces meeting at a single apex. Its volume is one third of the base area multiplied by the height.

In a coordinate plane, the length of the segment joining points (x₁, y₁) and (x₂, y₂), given by the square root of (x₂ − x₁)² + (y₂ − y₁)².

An algebraic description of a straight line in the plane, such as y = mx + b with slope m and intercept b, or the general form ax + by + c = 0.

The measure of steepness of a line, equal to the change in y divided by the change in x between two points. Parallel lines share it; perpendicular slopes multiply to −1.

The point exactly halfway between two endpoints, found by averaging their coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2).