Post on 13-Apr-2017
Introduction to set theory and to methodology and philosophy ofmathematics and computer programming
Ordered pairs
An overview
by Jan Plaza
c©2017 Jan PlazaUse under the Creative Commons Attribution 4.0 International License
Version of February 14, 2017
{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
-
r5
3〈5, 3〉
〈5, 3〉 6=〈3, 5〉
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}}
Proposition: 〈a, b〉 = 〈c, d〉 iff a = c and b = d .
(←) Obvious.(If the first object is known under the names a and c , and the secondobject is known under the names b and d , then 〈a, b〉 and 〈c , d〉 areordered pairs formed from the same objects, and they are the same.)
(→) on the next slide
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 secondobject is known under the names b and d , then 〈a, b〉 and 〈c , d〉 areordered pairs formed from the same objects, and they are the same.)
(→) on the next slide
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 secondobject is known under the names b and d , then 〈a, b〉 and 〈c , d〉 areordered pairs formed from the same objects, and they are the same.)
(→)
on the next slide
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 secondobject is known under the names b and d , then 〈a, b〉 and 〈c , d〉 areordered pairs formed from the same objects, and they are the same.)
(→)
on the next slide
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 secondobject is known under the names b and d , then 〈a, b〉 and 〈c , d〉 areordered pairs formed from the same objects, and they are the same.)
(→)
on the next slide
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 secondobject is known under the names b and d , then 〈a, b〉 and 〈c , d〉 areordered pairs formed from the same objects, and they are the same.)
(→) on the next slide
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.
〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.
〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.
〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.
〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.
〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.
So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.
a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.
So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.
So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.
So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.
So, {a} = {c}.So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.
So, a = c .So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .
So, b = d .
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 6=b.〈a, b〉 = 〈c , d〉.So, {{a}, {a, b}} = {{c}, {c , d}}.a 6=b.So, {a, b} 6={c}.So, {a, b} = {c , d}.So, {c , d} 6={a}.So, {a} = {c}.So, a = c .So, b = d .