Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Ordered tuples
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 25, 2017

2. An ordered n-tuple – a sequential arrangement of n of objects; order matters, the
same object may occur at different positions.
A point in 3-dimensional space is represented as an ordered 3-tuple/triple x, y, z .
In a relational database, data is stored in tables.
Name Account Balance
"Tom Smith" "A232323" 145.30
"Jane Brown" "A456711" 2340.88
A table is thought of as a set of rows, every row being an ordered tuple.
The table above is thought of as a set of 3-tuples/triples:
{ "Tom Smith", "A232323", 145.30 , "Jane Brown", "A456711", 2340.88 }

3. Definition
1. The (ordered) 1-tuple with coordinate x1 , denoted x1 , is equal to x1.
2. The (ordered) 2-tuple with coordinates x1, x2 , denoted x1, x2 , is the ordered
pair x1, x2 .
3. For n > 2, The (ordered) n-tuple with coordinates x1, ..., xn , denoted
x1, ..., xn , is x1, ..., xn−1 , xn .
a = a
a, b = {{a}, {a, b}}
a, b, c = a, b , c ={{ a, b }, { a, b , c}}={{{{a}, {a, b}}}, {{{a}, {a, b}}, c}}
Proposition
x1, ..., xn = y1, ..., yn iff x1 =y1 and ... and xn =yn.

4. Definition
Alternative names for n-tuples:
ordered 1-tuple – monad ,
ordered 2-tuple – (ordered) pair ,
ordered 3-tuple – (ordered) triple ,
ordered 4-tuple – (ordered) quadruple ,
ordered 5-tuple – (ordered) quintuple ,
ordered 6-tuple – (ordered) sextuple ,
ordered 7-tuple – (ordered) septuple ,
ordered 8-tuple – (ordered) octuple .
Instead of “n-tuple” we can also say just a tuple .