Understanding 0-ary Operations: Definition and Examples

In summary, a 0-ary operation, also known as a nullary operation, is a constant symbol that selects one element from a set. It does not require any inputs and returns a member of the set. In the example of fields, 0 and 1 are nullary operations and are also the identity elements for the binary operations of addition and multiplication. When defining a 0-ary operation, any value from the set can be chosen, but once it is defined, only that value can be assigned to 0.
  • #1
matheinste
1,068
0
Hello all.

Along the lines of unary and binary operations, could someone describe what 0-ary operation is and possibly give an example. I have only seen such an operation mentioned once, and that was early in the first chapter of Grillet's Abstract Algebra (2007), and so presumably it does not figure figure very highly, but is of some interest. The explanation in the above mentioned book was not very clear to me.

Thanks. Matheinste.
 
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  • #2
A 0-ary (or "nullary") operation is also known as a "constant symbol".

Example. The first-order logical theory of "fields" can be formulated in many ways.One of them has...
two binary operations:
[tex](x,y) \mapsto x+y[/tex]
[tex](x,y) \mapsto x\cdot y[/tex]
two unary operations:
[tex] x \mapsto -x[/tex]
[tex] x \mapsto 1/x[/tex]
and two nullary operations:
[tex]0[/tex]
[tex]1[/tex]
then some axioms that these operations must satisfy. Actually, [tex]1/x[/tex] is not defined for [tex]x=0[/tex], so some adjustment would have to be made for that.
 
  • #3
g_edgar said:
A 0-ary (or "nullary") operation is also known as a "constant symbol".

Example. The first-order logical theory of "fields" can be formulated in many ways.One of them has...
two binary operations:
[tex](x,y) \mapsto x+y[/tex]
[tex](x,y) \mapsto x\cdot y[/tex]
two unary operations:
[tex] x \mapsto -x[/tex]
[tex] x \mapsto 1/x[/tex]
and two nullary operations:
[tex]0[/tex]
[tex]1[/tex]
then some axioms that these operations must satisfy. Actually, [tex]1/x[/tex] is not defined for [tex]x=0[/tex], so some adjustment would have to be made for that.

Thanks for your reply.

In the case you have cited are the zero and one nullary operators the same as the identity elements for the two field operations?

Matheinste.
 
  • #4
matheinste said:
In the case you have cited are the zero and one nullary operators the same as the identity elements for the two field operations?

Yes. Among the axioms would be
[tex](\forall x)\; (x+0=x)[/tex]
[tex](\forall x)\; (x\cdot 1 = x)[/tex]
 
  • #5
g_edgar said:
Yes. Among the axioms would be
[tex](\forall x)\; (x+0=x)[/tex]
[tex](\forall x)\; (x\cdot 1 = x)[/tex]


But aren't these binary operations.

Matheinste.
 
  • #6
matheinste said:
But aren't these binary operations.

+ and * are, but 0 and 1 aren't.
 
  • #7
Moo Of Doom said:
+ and * are, but 0 and 1 aren't.

Yes, so they are.

My problem, mental block, is that a binary operation on a set is an operation needing two inputs from the set which returns a member of the set. A unary operation has a single input and returns a member of the set. And so a nullary operation has no inputs and returns a member of the set. Does this equate to just picking a member of the set.

Matheinste.
 
  • #8
A unary operation can be identified with its output, yes. That's why g_edgar wrote 0 and 1 instead of 0() and 1().
 
  • #9
Hello again

----A 0-ary or constant operation on a set S is a mapping f : {0} −→ S and simply selects one element f (0) of S. -----

Can anyone please explain what exactly does the "element f(0) of S" mean in the above sentence, from Grillet, Abstract Algebra, .

Matheinste.
 
  • #10
f(0) denotes the value assigned to 0 by f. The value will be a member of f's range, which is S.
 
  • #11
honestrosewater said:
f(0) denotes the value assigned to 0 by f. The value will be a member of f's range, which is S.

Does this leave us free to assign any member of the range, which will be a member of the set S, as the image of the function.

Matheinste.
 
  • #12
When you define the function, you can choose any value from S. But the requirement that a function assign only one value to each member of the domain means that you cannot assign any more values after one has been chosen. You would have to define a new function and give it a new name.
 

What are -ary set operations?

-ary set operations are mathematical operations that involve sets with a specific number of elements. The "-ary" prefix indicates the number of sets involved, such as binary (2 sets), ternary (3 sets), or n-ary (n sets).

What are the common types of -ary set operations?

The most common types of -ary set operations are union, intersection, difference, and complement. Union combines all elements from two or more sets, intersection finds the common elements between two or more sets, difference finds the elements in one set but not in another, and complement finds the elements not included in a particular set.

How are -ary set operations used in real life?

-ary set operations are used in various fields such as computer science, statistics, and engineering to analyze and manipulate data. They can be used to compare and combine data sets, find similarities or differences between groups, and identify patterns in data.

What are the properties of -ary set operations?

The properties of -ary set operations include commutativity, associativity, and distributivity. Commutativity means the order of sets does not affect the result, associativity means the grouping of sets does not affect the result, and distributivity means the operation can be distributed over other operations.

What are some examples of -ary set operations?

Some examples of -ary set operations include finding the union of two sets of numbers (binary), finding the intersection of three sets of names (ternary), finding the difference between four sets of colors (quaternary), and finding the complement of n sets of letters (n-ary).

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