Function Composition - Alternative Notations

Alternative Notations

• Many mathematicians omit the composition symbol, writing gf for g ∘ f.

• In the mid-20th century, some mathematicians decided that writing "g ∘ f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They write "xf" for "f(x)" and "(xf)g" for "g(f(x))". This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when x is a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notation is called postfix notation. The order is important because matrix multiplication is non-commutative. Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.

• Mathematicians who use postfix notation may write "fg", meaning first do f then do g, in keeping with the order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists may write "f;g" for this, thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation a fat semicolon ⨟ (U+2A1F) is used for left relation composition. Since all functions are binary relations, it is correct to use the fat semicolon for function composition as well (see the article on Composition of relations for further details on this notation).