Parentheses are overused

Parentheses are used to represent all sorts of operations and objects, many of which conflict with each other.

Grouping parts of an expression: \( (x+1)(x+2) \)

Argument of a function: \(f(x)\) is "\(f\) applied to \(x\)". (There is no function application symbol)

Greatest common divisor: \((a,b) = \gcd(a,b)\)

Counting combinations: \( {n \choose k} = \frac{n!}{k!(n-k)!} \)

Repeated differentiation : \( f^{(n)}(x) = \frac{\mathrm{d}^nf}{\mathrm{d}x^n} \)

Vectors or one-column matrices: \( \begin{pmatrix} a \\ b \end{pmatrix} \)

Ideals: \((2)\) is the ideal generated by 2, \((a,b,c)\) is the ideal generated by \(\{a,b,c\}\).

Tuples: \((a,b)\)

Cycle notation for permutations: \((a,b)\) or \((a \; b)\)

Legendre/Jacobi symbol: \( \left(\dfrac{a}{b}\right) \)

Intervals: \((a,b)\) is open at both ends, \((a,b]\) is open at one end.

Less standard notations, local to a document

These are many and varied, one motivating example :

The set of all 2-faces of the simplex with vertices a,b,c and d: \((a,b,c,d)_2\).