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2.9.15 Operators without Built-in Meanings

When you enter a piece of input such as 2 + 2, Mathematica first recognizes the + as an operator and constructs the expression Plus[2, 2], then uses the built-in rules for Plus to evaluate the expression and get the result 4.

But not all operators recognized by Mathematica are associated with functions that have built-in meanings. Mathematica also supports several hundred additional operators that can be used in constructing expressions, but for which no evaluation rules are initially defined.

You can use these operators as a way to build up your own notation within the Mathematica language.

The is recognized as an infix operator, but has no predefined value.

In[1]:=

Out[1]//FullForm=

In StandardForm, prints as an infix operator.

In[2]:=

Out[2]=

You can define a value for .

In[3]:= x_ CirclePlus y_ := Mod[x + y, 2]

Now is not only recognized as an operator, but can also be evaluated.

In[4]:= 2 CirclePlus 3

Out[4]=

A few Mathematica operators corresponding to functions without predefined values.

Mathematica follows the general convention that the function associated with a particular operator should have the same name as the special character that represents that operator.

\[Congruent] is displayed as .

In[5]:= x \[Congruent] y

Out[5]=

It corresponds to the function Congruent.

In[6]:= FullForm[%]

Out[6]//FullForm=

The conventional correspondence in Mathematica between operator names and function names.

You should realize that even though the functions CirclePlus and CircleTimes do not have built-in evaluation rules, the operators and do have built-in precedences. Section A.2.7 lists all the operators recognized by Mathematica, in order of their precedence.

The operators and have definite precedences—with higher than .

In[7]:= x CircleTimes y CirclePlus z // FullForm

Out[7]//FullForm=

Some two-dimensional forms without built-in meanings.

Subscripts have no built-in meaning in Mathematica.

In[8]:=

Out[8]//InputForm= Subscript[x, 2] + Subscript[y, 2]

Most superscripts are however interpreted as powers by default.

In[9]:=

Out[9]//InputForm= x^2 + y^2

A few special superscripts are not interpreted as powers.

In[10]:=

Out[10]//InputForm= SuperDagger[x] + SuperPlus[y]

Bar and hat are interpreted as OverBar and OverHat.

In[11]:=

Out[11]//InputForm= OverBar[x] + OverHat[y]