**2.8.15 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.
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Out[2]=

You can define a value for .
In[3]:= **x_ y_ := Mod[x + y, 2]**

Now is not only recognized as an operator, but can also be evaluated.
In[4]:= **2 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[%]**

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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. SectionÂ A.2.7 lists all the operators recognized by Mathematica, in order of their precedence.

The operators and have definite precedenceswith higher than .
In[7]:= **x y z // FullForm**

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Some two-dimensional forms without built-in meanings.
Subscripts have no built-in meaning in Mathematica.
In[8]:=

Out[8]//InputForm=

Most superscripts are however interpreted as powers by default.
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A few special superscripts are not interpreted as powers.
In[10]:=

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Bar and hat are interpreted as OverBar and OverHat.
In[11]:=

Out[11]//InputForm=