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 pairs
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 14, 2017

2. {x, y} – an unordered pair.
{x, y} = {y, x} – not so with ordered pairs.
A point on the plane is represented as an ordered pair of real numbers.
6
-
r
5
3
5, 3
5, 3 = 3, 5

3. Definition (Kuratowski)
The ordered pair with coordinates x, y , denoted x, y , is the set {{x}, {x, y}}
{x, y} tells that x and y are the components of the ordered pair.
{x} distinguishes the first component.
Fact
x, x = {{x}, {x, x}} = {{x}, {x}} = {{x}}

4. Proposition: a, b = c, d iff a = c and b = d.

5. Proposition: a, b = c, d iff a = c and b = d.
Proof

6. Proposition: a, b = c, d iff a = c and b = d.
Proof
(←)
(→)

7. Proposition: a, b = c, d iff a = c and b = d.
Proof
(←) Obvious.
(→)

8. Proposition: a, b = c, d iff a = c and b = d.
Proof
(←) Obvious.
(If the first object is known under the names a and c, and the second
object is known under the names b and d, then a, b and c, d are
ordered pairs formed from the same objects, and they are the same.)
(→)

9. Proposition: a, b = c, d iff a = c and b = d.
Proof
(←) Obvious.
(If the first object is known under the names a and c, and the second
object is known under the names b and d, then a, b and c, d are
ordered pairs formed from the same objects, and they are the same.)
(→) on the next slide

10. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→)

11. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .

12. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
Case: a=b.

13. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
Case: a=b.

14. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
Case: a=b.

15. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
Case: a=b.

16. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.

17. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .

18. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.

19. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.

20. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.
So, {a, b}={c}.

21. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.
So, {a, b}={c}.
So, {a, b} = {c, d}.

22. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.
So, {a, b}={c}.
So, {a, b} = {c, d}.
So, {c, d}={a}.

23. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.
So, {a, b}={c}.
So, {a, b} = {c, d}.
So, {c, d}={a}.
So, {a} = {c}.

24. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.
So, {a, b}={c}.
So, {a, b} = {c, d}.
So, {c, d}={a}.
So, {a} = {c}.
So, a = c.

25. Proposition: a, b = c, d iff a = c and b = d.
Proof
(→) Assume that a, b = c, d .
Case: a = b.
a, b = c, d .
So, {{a}} = {{c}, {c, d}}.
So, {a} = {c} = {c, d}.
So, a = c = d.
Case: a=b.
a, b = c, d .
So, {{a}, {a, b}} = {{c}, {c, d}}.
a=b.
So, {a, b}={c}.
So, {a, b} = {c, d}.
So, {c, d}={a}.
So, {a} = {c}.
So, a = c.
So, b = d.

26. Fact
x, y = y, x iff x = y.

27. Thinking about program variables and their values. (In two different ways.)
1. A variable has a value: VARIABLE-NAME, VALUE .
2. A variable refers to a memory address. The memory address stores a value.
VARIABLE-NAME, MEMORY-ADDRESS together with
MEMORY-ADDRESS, VALUE .