# Probability/Set Theory

## Introduction

The overview of set theory contained herein adopts a naive point of view. A rigorous analysis of the concept belongs to the foundations of mathematics and mathematical logic. Although we shall not initiate a study of these fields, the rules we follow in dealing with sets are derived from them.

## Sets

Definition.

(Set)

• A set is a well-defined collection of distinct object(s), which are called element(s).
• We say that an element belongs to the set.
• If $x$  belongs to a set $S$  , we write $x\in S$ .
• If $x$  does not belong to the set $S$  , we write $x\notin S$ .

Remark.

• When $x$  and $y$  are (not) equal, denoted by $x=(\neq )\;y$  , $x$  and $y$  are different symbols denoting the same (different) object(s).

Example. (Collection that is not well-defined)

• The collection of easy school subjects is not a set.

We have different ways to describe a set, e.g.

• word description: e.g., a set $S$  is the set containing the 12 months in a year;
• listing: elements in a set are listed within a pair of braces, e.g., $S{\overset {\text{ def }}{=}}\{{\text{January, March, February, April, May, June, July, August, September, October, November, December}}\}$ ;
the ordering of the elements is not important, i.e. even if the elements are listed in different order, the set is still the same. E.g., $\{{\text{January, February, March, April, May, June, July, August, September, October, November, December}}\}$  is still referring to the same set.
• set-builder notation:
$\underbrace {\{} _{{\text{The set of}}\;}\underbrace {x} _{{\text{all elements }}x\;}\underbrace {:} _{\text{such that }}\underbrace {P(x)} _{{\text{the property }}P(x){\text{ holds}}}\}$

in which the closing brace must also be written. E.g., $S{\overset {\text{ def }}{=}}\{x:x{\text{ is a month in a year}}\}$ .
• In particular, since a set contains distinct objects, the months contained in this set are distinct, and therefore there are only 12 elements in this set.

Example. (Empty set) The set $\{\}$  is called an empty set, and it contains no element. It is commonly denoted by $\varnothing$  also.

Exercise.

Is $\{\varnothing \}$  an empty set?

 Yes. No.

Example.

• ${\text{apple}}\in \{{\text{apple, orange, banana}}\}$ ;
• $\varnothing \in \{\varnothing \}$ ;
• $\varnothing \notin \varnothing$ .

Exercise.

Select all element(s) belonging to the set ${\big \{}\varnothing ,\{\varnothing \},\{\{\varnothing \}\}{\big \}}$ .

 $\varnothing$ $\{\varnothing \}$ $\{\varnothing ,\{\varnothing \}\}$ $\{\{\varnothing \}\}$ $\{\{\varnothing ,\{\varnothing \}\}\}$ Definition. (Set equality) When two sets are equal, they contain the same elements.

Remark.

• Equivalently, two sets $A$  and $B$  are equal if each element of $A$  is also element of $B$  and each element of $B$  is also element of $A$ .
• We use $A=(\neq )\;B$  to denote sets $A$  and $B$  are (not) equal.

Example.

• $\{\varnothing \}$ , $\{x:x{\text{ is a empty set}}\}$ , and the set that contains only an empty set are pairwise equal.
• $\varnothing =\{\}\neq {\big \{}\{\}{\big \}}=\{\varnothing \}$ .
• $\{2,3,5\}=\{x:x{\text{ is a prime number that is not greater than }}5\}$

Definition. (Universal set) Universal set, denoted by $U$ , is the set that contains all objects being considered in a particular context.

Remark.

• In the context of probability, a universal set, which is usually denoted by $\Omega$  instead, is the set containing all outcomes of a particular random experiment, and is also called a sample space.

Definition. (Cardinality) Cardinality of a finite set, which is a set containing finite number of elements, is the number of its elements.

Remark.

• Cardinality of set $S$  can be denoted by $\#(S)$  (or $|S|$ )
• We do not use the $|\cdot |$  notation to avoid ambiguity.
• Infinite set is a set containing infinite number of elements.
• We will leave the cardinality of infinite set undefined in this book, but it can be defined, in a more complicated way.

Example.

• $\#({\big \{}\{1\},2,3{\big \}})=3$ .
• $\#(\varnothing )=0$ .
• $\mathbb {N}$  (the set containing each positive number) is an infinite set.

Exercise.

Calculate $\#(\{\varnothing ,\{\varnothing \}\})$ .

 0 1 2 3 None of the above.

## Subsets

We introduce a relationship between sets in this section.

Definition. (Subset)

• If each element of set $A$  is an element of set $B$ , then $A$  is a subset of $B$ , denoted by $A\subseteq B$ .
• If $A$  is not a subset of $B$ , then we write $A\not \subseteq B$ .

Remark.

• By referring to the definitions of subsets and set equality, we can see that $A=B$  is equivalent to (or if and only if) $A\subseteq B$  and $B\subseteq A$ .
• The notation $A\supseteq B$  means that $A$  is a superset of $B$ , which means that $B$  is a subset of $A$ .
• This notation and terminology are seldom used.

Definition. (Venn diagram) A Venn diagram is a diagram that shows all possible logical relations between finitely many sets.

Remark.

• It is quite useful for illustrating some simple relationships between sets, and making the relationships clear.
• We may also add various annotations in the Venn digram, e.g. cardinality of each set, and the element(s) contained by each set.

Illustration of subset by Venn diagram:

A ⊆ B (A ≠ B):

*-----------------------*
|                       |
|                       |
|   *----------*        | <---- B
|   |          |        |
|   |    A     |        |
|   |          |        |
|   *----------*        |
*-----------------------*


Example.

• $\{1,3\}\subseteq \{1,2,3\}$ ;

Venn digram:

*-----------------------*
|                       |
|                       |
|   *----------*  2     |
|   |          |        |
|   |    1  3  |        |
|   |          |        |
|   *----------*        |
*-----------------------*

• $\{\{1\}\}\not \subseteq \{1,2,3\}$ ;

Venn diagram:

*-----------------------*
|                       |
|                       |
|   *----------*  {1}   |
|   |          |        |
|   |  1 2  3  |        |
|   |          |        |
|   *----------*        |
*-----------------------*

• $\varnothing \subseteq S$  for each set $S$  ;
• $S\subseteq S$  for each set $S$ .

Example. (Intervals) Intervals are commonly encountered subsets of $\mathbb {R}$ . If $a$  and $b$  are (extended) real numbers such that $a , then

{\begin{aligned}{\color {Maroon}(}a,b{\color {Maroon})}&{\overset {\text{ def }}{=}}\{x\in \mathbb {R} :a\;{\color {Maroon}<}\;x\;{\color {Maroon}<}\;b\};\\{\color {darkgreen}[}a,b{\color {Maroon})}&{\overset {\text{ def }}{=}}\{x\in \mathbb {R} :a\;{\color {darkgreen}\leq }\;x\;{\color {Maroon}<}\;b\};\\{\color {Maroon}(}a,b{\color {darkgreen}]}&{\overset {\text{ def }}{=}}\{x\in \mathbb {R} :a\;{\color {Maroon}<}\;x\;{\color {darkgreen}\leq }\;b\};\\{\color {darkgreen}[}a,b{\color {darkgreen}]}&{\overset {\text{ def }}{=}}\{x\in \mathbb {R} :a\;{\color {darkgreen}\leq }\;x\;{\color {darkgreen}\leq }\;b\}.\\\end{aligned}}

In particular, $(-\infty ,\infty )=\mathbb {R}$ , and $[-\infty ,\infty ]$  is the set containing all extended real numbers.

Definition. (Proper subset)

• Set $A$  is a proper subset of set $B$  if $A\subseteq B$  and $A\neq B$ ;. We write $AsubsetneqB$  in this case.
• If set $A$  is not a proper subset of $B$ , then we write $A\not \subsetneq B$  (but we rarely write this).

Remark.

• The notation $A\supsetneq$  means that $A$  is a proper superset of $B$ , which means that $B$  is a proper subset of $A$ .
• This notation and terminology are seldom used.

Example.

• $\{1,2\}\not \subsetneq \{1,2\}$ .

Definition. (Complement) Let $A$  be a subset of universal set $U$ . The (absolute) complement of $A$ , denoted by $A^{c}$ , is the set $\{x\in U:x\notin A\}$ .

Example. If $A=\{1,2,3\}$  and $U=\{1,2,3,4,5\}$ , then $A^{c}=\{4,5\}$ .

Venn diagram:

*-----------------------*
|                       |
|                4  5   |
|   *----------*        |
|   |          |        | <---- U
|   |  1 2  3  |        |
|   |          |        |
|   *----------*        |
*--------^--------------*
|
A


Exercise.

Find $\varnothing ^{c}$ .

 $U$ $\{U\}$ $\varnothing$ $\{\varnothing \}$ None of the above.

## Set operations

Probability theory makes extensive use of some set operations, and we will discuss them in this section.

Definition.

(Union of sets) Union of set $A$  and set $B$ , denoted by $A\cup B$ , is the set $\{x:x\in A{\text{ or }}x\in B\}$ .

Remark.

• $A\cup B$  is read 'A cup B'.
• We can denote $A_{1}\cup A_{2}\cup \dotsb$  by $\bigcup _{i=1}^{n}A_{i}$  (if the sequence of unions stops at $A_{n}$ ), or $\bigcup _{i=1}^{\infty }A_{i}$  (if the sequence of unions does not stop).

Example.

• $\{{\text{apple}},{\text{orange}}\}\cup \{{\text{orange}},{\text{red}}\}=\{{\text{apple}},{\text{orange}},{\text{red}}\}$ .

Venn diagram:

*----------------*
|                |
|  red   *-------*--------*
|        | orange|        |
*--------*-------*        |
|       apple    |
*----------------*


Proposition. (Properties of union of sets) Let $A$ ,$B$  and $C$  be sets. Then, the following statements hold.

(a) $A\cup A=A$ ;
(b) $A\cup B=B\cup A$  (commutative law);
(c) $A\cup (B\cup C)=(A\cup B)\cup C$  (associative law);
(d) $A\subseteq A\cup B$  and $B\subseteq A\cup B$ ;
(e) $A\cup \varnothing =A$ ;
(f) $A\cup B=B$  if and only if $A\subseteq B$ .

Proof. Informally, consider the following Venn diagrams:

(a)
*----*
|    | <---- A ∪ A (a set overlaps itself completely)
|    | <--- A
*----*

(b)
*----------------*
|////////////////| <---- A
|////////*-------*--------*
|////////|///////|////////|
*--------*-------*////////| <--- B
|////////////////|
*----------------*
(shaded region refers to both A ∪ B and B ∪ A)

(c)
*----------*
|//////////| <--- A
|//////////|
|/////*----|----*
|/////|////|////| <---- C
*-----*----*----*------------*
|/////|////|////|////////////| <--- B
|/////*----|----*////////////|
*----------*-----------------*
(shaded region refers to both A∪(B∪C) and (A∪B)∪C)

(d)
*----------------*
|                | <---- A
|        *-------*--------*
|        |       |        |
*--------*-------*        | <--- B
|                |
*----------------*

$A$  and $B$  are both inside the whole region, which represents $A\cup B$ .
(e)
*----*
|    |
|    | .
*----*

(f)
*-----------------------*
|                       |
|                       |
|   *----------*        | <---- B
|   |          |        |
|   |    A     |        |
|   |          |        |
|   *----------*        |
*-----------------------*
(the whole region is both A u B and B)

We use a dot, which has zero area, to represent $\varnothing$ . Then, we can see that the union of the region and the dot is the region itself.

$\Box$

Remark.

• Formal proofs of propositions and theorems about sets will not be emphasized in this book.
• Instead, we will usually prove the propositions and theorems informally, e.g. using Venn diagram.

Definition. (Intersection of sets)

Intersection of set $A$  and set $B$ , denoted by $A\cap B$ , is the set $\{x:x\in A{\text{ and }}x\in B\}$ .

Remark.

• $A\cap B$  is read 'A cap B'.
• We can denote $A_{1}\cap A_{2}\cap \dotsb$  by $\bigcap _{i=1}^{n}A_{i}$  (if the sequence of intersections stops at $A_{n}$ ), or $\bigcap _{i=1}^{\infty }A_{i}$  (if the sequence of intersection does not stop).

Example.

• $\{1,2,3\}\cap \{2,3,4\}=\{2,3\}$ ;
• $\{1,2,3\}\cap \{4,5,6\}=\varnothing$ .

Definition. (Disjoint sets) Set $A$  and set $B$  are disjoint (or mutually exclusive) if $A\cap B=\varnothing$ .

Remark.

• I.e., $A$  and $B$  are disjoint if they have no element in common.
• More than two events are disjoint if they are pairwise disjoint.

Venn diagram

*-----*       *-----*       *-----*
|     |       |     |       |     |
|  A  |       |  B  |       |  C  |
*-----*       *-----*       *-----*

(A, B and C are disjoint)

*----------------*
|                | <---- D
| *--*   *-------*--------*
| |  |   |       |        |
*-*--*---*-------*        | <--- E
|  |   |                |
*--*   *----------------*
^
|
F

(D, E and F are not disjoint, but E and F are disjoint)


Definition. (Partition of a set) A collection of sets form a partition of a set $S$  if the sets in the collection are disjoint and their union is $S$ .

Venn diagram

*-----------------------*
| \        A            |
|  \                    |
|B  *-------------------*
|   /                   |
|  /       C            |
| /         *-----------*  <----- S
|/         /            |
*\        /             |
| *------*              |
|        |    E         |
|  D     |              |
*--------*--------------*

(A,B,C,D and E form a partition of S)


Proposition. (Properties of intersection of sets) Let $A$ ,$B$  and $C$  be sets. Then, the following statements hold.

(a) $A\cap A=A$ ;
(b) $A\cap B=B\cap A$  (commutative law);
(c) $A\cap (B\cap C)=(A\cap B)\cap C$  (associative law);
(d) $A\cap B\subseteq A$  and $A\cap B\subseteq B$ ;
(e) $A\cap \varnothing =\varnothing$ ;
(f) $A\cap B=B$  if and only if $B\subseteq A$ .

Proof. Informally, consider the following Venn diagrams:

(a)
*----*
|    | <---- A ∩ A (a set overlaps itself completely)
|    | <--- A
*----*

(b)
*----------------*
|                | <---- A
|        *-------*--------*
|        |A∩B=B∩A|        |
*--------*-------*        | <--- B
|                |
*----------------*

(c)
*----------*
|          | <--- A
|          |
|     *----*----*
|     |    |    | <---- C
*-----*----*----*------------*
|     |////|    |            | <--- B
|     *----*----*            |
|          |                 |
*----------*-----------------*

*----*
|////| : A∩(B∩C)=(A∩B)∩C
*----*

(d)
*----------------*
|                | <---- A
|        *-------*--------*
|        | A ∩ B |        |
*--------*-------*        | <--- B
|                |
*----------------*
(A ∩ B is inside A, and A ∩ B is inside B)

(e)
*----*
|    |
| .  |
*----*

We use a dot, which has zero area, to represent $\varnothing$ . Then, we can see that the intersection of the region and the dot is the dot.

$\Box$

Proposition. (Distributive law) Let $A$ ,$B$  and $C$  be sets. Then, the following statements hold.

(a) $A\cap ({\color {darkgreen}B}\cup {\color {maroon}C})=(A\cap {\color {darkgreen}B})\cup (A\cap {\color {maroon}C})$ ;
(b) $A\cup ({\color {darkgreen}B}\cap {\color {maroon}C})=(A\cup {\color {darkgreen}B})\cap (A\cup {\color {maroon}C})$ .

Proof.

(a)
*----------*
|          | <--- A
|          |
|     *----*----*
|     |    |    | <---- C
*-----*----*----*------------*
|/////|////|    |            | <--- B
|/////*----*----*            |
|//////////|                 |
*----------*-----------------*

*----*
|////| : AnB
*----*

*----------*
|          | <--- A
|          |
|     *----*----*
|     |////|    | <---- C
*-----*----*----*------------*
|     |////|    |            | <--- B
|     *----*----*            |
|          |                 |
*----------*-----------------*

*----*
|////| : AnC
*----*

*----------*
|          | <--- A
|          |
|     *----*----*
|     |////|    | <---- C
*-----*----*----*------------*
|/////|////|    |            | <--- B
|/////*----*----*            |
|//////////|                 |
*----------*-----------------*

*----*
|////| : (AnB)u(AnC)
*----*

*----------*
|          | <--- A
|          |
|     *----*----*
|     |////|////| <---- C
*-----*----*----*------------*
|/////|////|////|////////////| <--- B
|/////*----*----*////////////|
|//////////|/////////////////|
*----------*-----------------*

*----*
|////| : BuC
*----*

*----------*
|          | <--- A
|          |
|     *----*----*
|     |////|    | <---- C
*-----*----*----*------------*
|/////|////|    |            | <--- B
|/////*----*----*            |
|//////////|                 |
*----------*-----------------*

*----*
|////| : An(BuC)
*----*

(b)
*----------*
|//////////| <--- A
|//////////|
|/////*----*----*
|/////|////|    | <---- C
*-----*----*----*------------*
|/////|////|////|////////////| <--- B
|/////*----*----*////////////|
|//////////|/////////////////|
*----------*-----------------*

*----*
|////| : AuB
*----*

*----------*
|//////////| <--- A
|//////////|
|/////*----*----*
|/////|////|////| <---- C
*-----*----*----*------------*
|/////|////|////|            | <--- B
|/////*----*----*            |
|//////////|                 |
*----------*-----------------*

*----*
|////| : AuC
*----*

*----------*
|//////////| <--- A
|//////////|
|/////*----*----*
|/////|////|    | <---- C
*-----*----*----*------------*
|/////|////|////|            | <--- B
|/////*----*----*            |
|//////////|                 |
*----------*-----------------*

*----*
|////| : (AuB)n(AuC)
*----*

*----------*
|          | <--- A
|          |
|     *----*----*
|     |    |    | <---- C
*-----*----*----*------------*
|     |////|////|            | <--- B
|     *----*----*            |
|          |                 |
*----------*-----------------*

*----*
|////| : B∩C
*----*

*----------*
|//////////| <--- A
|//////////|
|/////*----*----*
|/////|////|    | <---- C
*-----*----*----*------------*
|/////|////|////|            | <--- B
|/////*----*----*            |
|//////////|                 |
*----------*-----------------*

*----*
|////| : Au(B∩C)
*----*


$\Box$

Definition. (Relative complement)

Relative complement of $A$  (left) in $B$  (right).

Relative complement of set $A$  in set $B$ , denoted by $B\setminus A$ , is the set $\{x:x\in B{\text{ and }}x\notin A\}$ .

Remark.

• If $U$  is the universal set and $A$  is a subset of $U$ , then $A^{c}=U\setminus A$ .
• $B\setminus A$  is read 'B minus A'.

Example.

• $\{1,2,3\}\setminus \{1,2\}=\{3\}$ ;
• $\{1,2,3\}\setminus \{1,2,3\}=\varnothing$ ;
• $\{1,2,3\}\setminus \{4,5,6\}=\{1,2,3\}$ .

Proposition. (Properties of relative complement) Let $A$  and $B$  be sets. Then, the following statements hold.

(a) $A\setminus A=\varnothing$ ;
(b) $A\setminus \varnothing =A$ ;
(c) $\varnothing \setminus A=\varnothing$ ;
(d) $A\setminus B=\varnothing$  if and only if $A\subseteq B$ ;
(e) $(A\setminus B)\cap (B\setminus A)=\varnothing$ ;
(f) $A\cap (B\setminus A)=\varnothing$ .

Proof.

• $A\setminus B$  can be viewed as 'removing' the region of $B$  from the region of $A$ .
(a)
*--*
|A |  removing the whole region <=> empty region left
*--*

(b)
*--*
|A |  removing empty region <=> whole region left
*--*

(c)
.  removing anything from an empty region <=> still an empty region

(d)
*----*
|    | <-- B
*--* |
|A | |    removing B from A becomes empty region <=> region B is not smaller than A
*--*-*

(e)
*-----*
| A\B |
*-----*-----*
|     | B\A | <--- B
*-----*-----*
^
|
A
(A\B and B\A are always disjoint)

(f)
*-----*
|     |
*-----*-----*
|     | B\A | <--- B
*-----*-----*
^
|
A

(A and B\A are always disjoint)


$\Box$

Theorem. (De Morgan's laws) Let $B,A_{1},A_{2},\dotsc$  be sets. Then,

$B\setminus (A_{1}\cup A_{2}\cup \dotsb )=(B\setminus A_{1})\cap (B\setminus A_{2})\cap \dotsb {\text{ and }}B\setminus (A_{1}\cap A_{2}\cap \dotsb )=(B\setminus A_{1})\cup (B\setminus A_{2})\cup \dotsb$

Proof.

(only 3 sets involved)
*-------------------------------*
|                    IV         |
|   *---------*                 |
|   |    I    | <--- A_1        | <--- B
|   *---------*-------*         |
|   |    II   | III   |<--- A_2 |
|   *---------*-------*         |
*-------------------------------*
B\(A_1uA_2)=IV
(B\A_1)=III u IV \
----> intersection: IV
(B\A_2)=I u IV   /

B\(A_1nA_2)=I u III u IV
(B\A_1)=III u IV \
----> union: I u III u IV
(B\A_2)=I u IV   /


$\Box$

Remark.

• If $B=U$ , then the equations become $(A_{1}\cup A_{2}\cup \dotsb )^{c}=A_{1}^{c}\cap A_{2}^{c}\cap \dotsb {\text{ and }}(A_{1}\cap A_{2}\cap \dotsb )^{c}=A_{1}^{c}\cup A_{2}^{c}\cup \dotsb$ .

Example. Let $A=\{1,2,3\},B=\{1,3\},C=\{1,2,3,4\}$  and $U=\{1,2,3,4,5\}$ . Then,

• $A\setminus (\underbrace {B\cup C} _{=C})=\varnothing =(A\setminus B)\cap (\underbrace {A\setminus C} _{=\varnothing })$ ;
• $\overbrace {C\setminus (\underbrace {A\cap B} _{=\{1,3\}})} ^{\{2,4\}}=(\underbrace {C\setminus A} _{=\{4\}})\cup (\underbrace {C\setminus B} _{=\{2,4\}})$ ;
*--------------------------*
|   *-------------------*  |
| 5 |///////////////////|  |
|   |/4/*------------*//|  |
|   |///|//////2/////|//|  |
|   |///|/*-------*//|//|  |
|   |///|/| 1    3|//|//|  |
|   |///|/|  AnB  |//|//| <-------- C
|   |///|/*-------*//|//|  |
|   |///|////////////|//|  |
|   |///*------------*//|  |
|   |///////////////////|  |
|   *-------------------*  |
*--------------------------*

*---*
|///| : C\(AnB)
*---*

*--------------------------*
|   *-------------------*  |
| 5 |\.\.\.\.\.\.\.\.\.\|  |
|   |\4\*------------*\.|  |
|   |\.\|.A\B..2.....|\.|  |
|   |\.\|.*-------*..|\.|  |
|   |\.\|.| 1    3|..|\.|  |
|   |\.\|.|   B   |..|\.| <-------- C
|   |\.\|.*-------*..|\.|  |
|   |\.\|............|\.|  |
|   |\.\*------------*\.|  |
|   |\.\\.\.\.\.\.\.\.\.|  |
|   *-------------------*  |
*--------------------------*

*---*
|...| : C\B
*---*
*---*
|\\\| : C\A
*---*

• $\overbrace {(\underbrace {A\cup B\cup C} _{=C})^{c}} ^{=\{5\}}=\underbrace {A^{c}} _{=\{4,5\}}\cap \underbrace {B^{c}} _{=\{2,4,5\}}\cap \underbrace {C^{c}} _{=\{5\}}$ .

Definition. (Power set) Power set ${\mathcal {P}}(A)$  (or $2^{A}$ ) of set $A$  is the set of all subsets of $A$ , i.e., $\{S:S\subseteq A\}$ .

Example.

• ${\mathcal {P}}(\{1,2\})=\{\varnothing ,\{1\},\{2\},\{1,2\}\}$ ;
• ${\mathcal {P}}(\varnothing )=\{\varnothing \}$  (power set of an empty set is not an empty set).

Remark.

• Power set of a set containing $n$  elements contains $2^{n}$  elements.

Definition. ($n$ -ary Cartesian product) The $n$ -ary Cartesian product over $n$  sets $S_{1},\dotsc ,S_{n}$ , denoted by $S_{1}\times \dotsb \times S_{n}$ , is

${\big \{}s_{1},\dotsc ,s_{n}:s_{i}\in S_{i}{\text{ for each }}i\in \{1,\dotsc ,n\}{\big \}}.$

Example. Let $A=\{1,2,3\},B=\{2,3,4\}$  and $C=\{3,4,5\}$ . Then,

• $A\times A=\{(1,1),(2,2),(3,3)\}$ ;
• $A\times B=\{(1,2),(2,3),(3,4)\}$ ;
• $A\times B\times C=\{(1,2,3),(2,3,4),(3,4,5)\}$ .

1. Extended real number system is obtained by adding $\infty$  and $-\infty$  to the real number system