Math notation reference

Operators

The following list gives the operators for the math notation definition in order of diminishing precedence. Any short forms for operators are given after the operator name. In definitions, A and B refer to the elements preceding and following the operator, respectively. Each precedence level has its own associativity.

1. Name Composition (left)

   index ;

   attach a quantitative index B to a variable name A
   example: x;1 + x;2 + 'ellipsis + x;n

   qual :

   attach a descriptive qualifier B to a variable name A
   example: V:be 'approxeq 0.7*'_Volt

2. Relation (right)

   apply (

   apply a relation A to an argument or tuple of arguments B

3. Exponentiation (right)

   exp ** ^

   raise a base A to an exponent B
   example: 'exp('exp(root(2),root(2)),root(2)) = 2

4. Factorial (unary left)

   fact !

   form the factorial of A

5. Multiplication and Division (left)

   prod *

   multiply two expressions
   example: 2 * 2 = 4

   quot /

   form the quotient of two expressions
   example: 'quot(a / b - 3,c)

   divby

   divide two expressions and suggest the dot-bar-dot symbol for visual rendering
   example: 6 'divby 2 = 3

   compose

   compose the function A with the function B

   isect

   form the intersection of two sets

6. Quantitative Negation (unary right)

   neg _

   form the negative of a quantity, vector, or matrix

7. Cross Product (left)

   cross

   form the cross product of two vectors

8. Dot Product (nonassoc)

   dot

   form the dot product of two vectors

9. Addition and Subtraction (left)

   sum +

   add two expressions

   diff -

   subtract two expressions

   union

   form the union of two sets

   disjunion

   form the disjoint union of two sets

   relcomp \

   form the relative complement of B in the set A

   symdiff

   form the symmetric difference of two sets

10. Quantitative Comparison (chain)

    eq =

    assert that two expressions are equal

    noteq

    assert that two expressions are not equal

    gt >

    assert that A is strictly greater than B

    gteq >=

    assert that A is greater than or equal to B

    lt <

    assert that A is strictly less than B

    lteq <=

    assert that A is less than or equal to B

    approxeq

    assert that A is approximately equal to B

    propto

    assert that A is proportional to B

    approach

    suppose that A approaches the value B

11. Geometric Relationships (chain)

    parallel ||

    assert that A is parallel to B

    perpto |_

    assert that A is perpendicular to B

    congruent =~

    assert that A is congruent to B

    similar ~~

    assert that A is similar to B

12. Set Comparison (chain)

    superseteq

    assert that A is a superset of the set B

    superset

    assert that A is a strict superset of the set B

    subseteq

    assert that A is a subset of the set B

    subset

    assert that A is a strict subset of the set B

    notsubset

    assert that A is not a subset of the set B

13. Containment (nonassoc)

    eltof

    assert that A is an element of the set B

    noteltof

    assert that A is not an element of the set B

14. Logical Negation (unary right)

    not

    assert the negation of a proposition or truth value

15. Logical Conjunction (left)

    and

    form the logical conjunction of two propositions or truth values

16. Logical Disjunction (left)

    or

    form the logical inclusive disjunction of two propositions or truth values

    xor

    form the logical exclusive disjunction of two propositions or truth values

17. Implication (serial)

    implies

    assert that proposition A implies proposition B

18. Logical Equivalence (serial)

    equiv

    assert that two propositions or truth values are equivalent

19. Definition (serial)

    defeq

    define A as being equal to B

20. Composition (serial)

    tuple ,

    join things into ordered lists

Please note a few things about operators:

• Multiplication must be explicitly indicated. Always use the * or times operator; never simply juxtapose two expressions.
• The apply operator and the tuple operator work together to allow natural notation such as f(x,y) to signify the application of a function to arguments.

  When, in mathematics, we commonly write f(x), we are establishing a relationship between f and x, indicating that f is to be treated as a function and applied to an argument x. This relationship is represented in MINSE by the apply compound, which has the function as its first sub-element and the argument as the second. When there are multiple arguments, they are first combined into a single compound by the comma operator, which builds a tuple compound out of all the arguments, and the tuple is used as the second sub-element of the apply compound.

  So beware: this use gives the parentheses and the comma a completely different meaning. Usually, the parentheses group the sub-elements of a compound and the commas separate these sub-elements. When not used after a quote and a compound's name, the parentheses act as the right-associative apply operator and commas act as the serial-associative tuple operator, performing the analogous (but distinct) roles of grouping and separating arguments to a mathematical function instead of the sub-elements of a MINSE compound. The function application becomes itself a compound, and likewise for the arguments if there are more than one.

  This is all done to reduce typing for the mathematician who is used to simply writing f(x,y) to represent a function of two arguments.