Operators

See also Operator Precedence Rules.

Special Symbols

Symbol Description Example
:= assignment
means "is assigned the value"		 >  x := 5;
>  y := 3*x-5;
>  S := {1, 9, 2, 4};
>  T := [3.2, 5, -4.1];
.. iteration >  S := {1..5};
>  S;
{2,1,3,4,5};
>  T := [1..5];
>  T;
[1,2,3,4,5]
| means "such that"
used in Set Formers and Tuple Formers		 >  S := {1..7};
>  S2 := {x | x in S | x > 5};
>  S2;
{7,6};
>  T := [1..7];
>  T2 := [x | x in T | x > 5];
>  T2;
[6,7];
: same as "|"
used in Set Formers and Tuple Formers		 See Above

Arithmetic Operators

Operator Description Example
+ Addition >  8713+246;
8959;
- Subtraction >  593.7-12.25;
581.450;
* multiplication >  123456789012345678901 * 12345;
1524074060357407406032845;
/ division (real division of reals and/or integers) >  47/2;
23.500;
div integer division >  47 div 2;
23;
>  46.3 div 23;
!Error: Bad arguments in:
46.300 div 23;
mod remainder from integer division >  12 mod 5;
2;
** exponeniation (raise to a power) >  2.5**4;;
39.063;

Relational Operators

Operator Description Example
= equal >  15.3/2 = 7.65;
true;
/= not equal >  15.3/2 /= 7.65;
false;
< less than >  2**5 < 30;
false;
> greater than >  2**5 > 30;
true;
<= less than or equal >  2**5 < = 32;
true;
>= greater than or equal >  2**5 > = 32;
true;

Logical or Propositional Operators

Operator Description Example
and true if both expressions are true; false if either or both are false >  (5>4) and(7<9);
true;
>  (5>4) and(7>9);
false;
>  (5<4) and(7<9);
false;
>  (5<4) and(7>9);
false;
or true if either or both expressions are true; false only if both are false >  (5>4) or(7<9);
true;
>  (5>4) or(7>9);
true;
>  (5<4) or(7<9);
true;
>  (5<4) or(7>9);
false;
not true if expression is false; false if expression is true >  not(5>4);
false;
>  not(5<4);
true;
impl logical implication. false if left expression is true and second is false; otherwise true >  (5>4) impl(7<9);
true;
>  (5>4) impl(7>9);
false;
>  (5<4) impl(7<9);
true;
>  (5<4) impl(7>9);
true;
= logical equivalence. true if both expressions are true or both false; false if one expression is true and other is false >  (5>4) = (7<9);
true;
>  (5>4) = (7>9);
false;
>  (5<4) = (7<9);
false;
>  (5<4) = (7>9);
true;
iff if and only if (same as logical equivalence) >  (5>4) = (7<9);
true;
>  (5>4) = (7>9);
false;
>  (5<4) = (7<9);
false;
>  (5<4) = (7>9);
true;

Predicate Operators

Operator Description Example
forall true if condition is true for all values in the set or tuple (Universal Quantifier) >  forall x in {1..5} | x < 5;
false;
>  forall x in [1..5] | x < 5;
false;
exists true if condition is true for at least one value in the set or tuple (Existential Quantifier) >  exists x in {1..5} | x < 5;
true;
>  exists x in [1..5] | x < 5;
true;
choose returns a value from the set or tuple for which the condition is true or OM if there are none for which it's true >  choose x in {1..5} | x < 5;
2;
>  choose x in [1..5] | x < 5;
4;

Set Operators

Operator Description Example
union
+		 set of all elements that are in either or both sets >  {1..5} union {3,4};
{2,1,5,4,3};
inter
*		 set of all elements that are in both sets >  {1..5} inter {3,4};
{4,3};
- set of all elements in the first set that are not in the second set >  {1..5} - {3,4};
{2,1,5};
= true if all elements of first set are in second set and all elements of the second set are in the first >  {1..5} = {3,4};
false;
/= true if either set contains at least one element that is not in the other set >  {1..5} /= {3,4};
true;
in

notin		 "in" is true if the given element is in the set

"notin" is true if the given element is not in the set		 >  x := 3;
>  S := {2,3,5,7,11};
>  x in S;
true;
>  x notin S;
false;
arb returns an arbitrary element of the set >  S := {2,3,5,7,11};
>  arb(S);
3;
>  arb(S);
11;
forall
exists
choose		 See Above (Predicate Operators)  
# number of elements in the set >  S := {2,3,5,7,11};
>  #(S);
5;
% causes a binary operation to be performed on all elements of the set. e.g. %+ means add up the elements, %* means multiply them all together >  %+{1..5};
15;
>  5%+{1..5};
20;
>  %*{1..5};
120;
>  5%*{1..5};
600;
with insert the given element into the set >  S := {1..5};
>  x := 17;
>  S2 := S with x;
>  S2;
{3,4,17,5,1,2};
less remove the given element from the set >  S := {1..5};
>  x := 4;
>  S2 := S less x;
>  S2;
{3,5,1,2};
take from remove an arbitrary element from the set and assign it to the given variable >  S := {1..5};
take x from S;
>  S;
{1,3,5,4};
>  x;
2;
subset true if first set is a subset of second set >  S1 := {1..5};
>  S2 := {3,5};
>  S3 := {};
>  S1 subset S2;
false;
>  S2 subset S1;
true;
>  S3 subset S1;
true;

Tuple Operators

Operator Description Example
+ concatenates two tuples >  [1..5]+[3,4];
[1,2,3,4,5,3,4];
* repeats the components of the tuple >  3*[3,4];
[3,4,3,4,3,4];
= true if two tuples have same number of components and every component of the first tuple is in the second at the same index >  [1..5] = [3,4];
false;
/= true if two tuples have different number of components or if some component of the first tuple is not in the second tuple at the same index >  [1..5] /= [3,4];
true;
in

notin		 "in" is true if the given element is in the tuple

"notin" is true if the given element is not in the tuple		 >  x := 3;
>  S := [2,3,5,7,11];
>  x in S;
true;
>  x notin S;
false;
arb returns an arbitrary component of the tuple >  T := [2,3,5,7,11];
>  arb(T);
7;
>  arb(T);
2;
forall
exists
choose		 See Above (Predicate Operators)  
# index of the last non-OM component

if all components are "defined," then it's the number of components		 >  T := [2,3,5,7,11];
>  #(T);
5;
>  T := [2,OM,5,7,OM];
>  #(T);
4;
% causes a binary operation to be performed on all components of the tuple. e.g. %+ means add up the components, %* means multiply them all together >  %+[1..5];
15;
>  5%+[1..5];
20;
>  %*[1..5];
120;
>  5%*[1..5];
600;
with insert the given component into the tuple at the end >  T := [1..5];
>  x := 17;
>  T2 := T with x;
>  T2;
[1,2,3,4,5,17};
take fromb remove one component from the beginning of the tuple and assign it to the given variable >  T := [1..5];
take x from T;
>  T;
[2,3,4,5];
>  x;
1;
take from same as take fromb See Above
take frome remove one component from the end of the tuple and assign it to the given variable >  T := [1..5];
take x from T;
>  T;
[1,2,3,4];
>  x;
5;