2.7 Ordered pairs

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


Proof




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


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


Proof


(→)

on the next slide


Proof



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 .


(→) 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 .


(→) 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 .


(→) 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.



(→) 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.



(→) 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.



(→) 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.



(→) 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 .


(→) 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 .


(→) 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 .


(→) 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 .


(→) 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 .


(→) 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 .


(→) 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 .


(→) 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 .


(→) 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 .

Fact〈x , y〉 = 〈y , x〉 iff x = y .

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〉.

Education

2.7 Ordered pairs