the parentheses around the argument altogether (e.g., ). Coordinates and vectors In the Cartesian coordinate system, brackets are used to specify the coordinates of a point. For example, (2,3) denotes the point with x-coordinate 2 and y-coordinate 3. The inner product of two vectors is commonly written as , but the notation (a, b) is also used. Intervals Both parentheses, ( ), and square brackets, [ ], can also be used to denote an interval. The notation is used to indicate an interval from a to c that is inclusive of —but exclusive of . That is, would be the set of all real numbers between 5 and 12, including 5 but not 12. Here, the numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In some European countries, the notation is also used for this, and wherever comma is used as decimal separator, semicolon might be used as a separator to avoid ambiguity (e.g., ). The endpoint adjoining the square bracket is known as closed, while the endpoint adjoining the parenthesis is known as open. If both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever infinity or negative infinity is used as an endpoint (in the case of intervals on the real number line), it is always considered open and adjoined to a parenthesis. The endpoint can be closed when considering intervals on the extended real number line. A common convention in discrete mathematics is to define as the set of positive integer numbers less or equal than . That is, would correspond to the set . Sets and groups Braces { } are used to identify the elements of a set. For example, {a,b,c} denotes a set of three elements a, b and c. Angle brackets ⟨ ⟩ are used in group theory and commutative algebra to specify group presentations, and to denote the subgroup or ideal generated by a collection of elements. Matrices An explicitly given matrix is commonly written between large round or square brackets: Derivatives The notation stands for the n-th derivative of function f, applied to argument x. So, for example, if , then . This is to be contrasted with , the n-fold application of f to argument x. Falling and rising factorial The