**Relations **

**Ex.**^{ }**The following are some binary relations on {1, 2, 3, 4}. **

**R**_{1}** = {(1, 1), (2, 2), (3, 3), (4, 4)}. **

**R**_{2}** = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}. **

**R**_{3}** = {(1, 2), (2, 1), (3, 4), (4, 3)}. **

**R**_{4}** = {(1, 2), (1, 3), (2, 3), (3, 4)}. **

**R**_{5}** = {(1, 1), (2, 2), (2, 3), (3, 4), (2, 4)}. **

**reflexive****: R**_{1}**, R**_{2 }**irreflexive****: R**_{3}**, R**_{4}** symmetric****: R**_{1}**, R**_{2}**, R**_{3}** asymmetric****: R**_{4}

** antisymmetric****: R**_{1}**, R**_{4}**, R**_{5 }

** transitive****: R**_{1}**, R**_{2}**, R**_{5 }

**Ex. Define R to be a binary relation on the set of integers, **** where *** a R b iff ab*≥

**0.**

**R is reflexive and symmetric. **

**R is not transitive (for example,**** ^{ }**−

**5 R 0,**

^{ }

**0 R 8,****but**−

**5**

**8).**

**Suppose that A****=****{1, 2, …, n}. **

**1. There are** **2**^{n}^{2}^{−}^{n}**reflexive binary relations on A. **

**Each reflexive binary relation on A must contain **** (1, 1), (2, 2), …, (n, n). **

* |A*×

**A − {(i, i)****: 1**≤

*≤*

**i**

**n}| = n****−**

^{2 }

**n.****2. There are** **2**^{(}^{n}^{2}^{+}^{n}^{)}^{/}^{2}**symmetric binary relations on A. **

* A*×

**A = {(i, i)****: 1**≤

*≤*

**i**

**n} + {(i, j)****: 1**≤

*≤*

**i***≤*

**n, 1***≤*

**j***≠*

**n, and i**

**j}.**** 2**^{n}** ×** **2**^{(}^{n}^{2}^{−}^{n}^{)}^{/}^{2}**=** **2**^{(}^{n}^{2}^{+}^{n}^{)}^{/}** ^{2}**.

**3. There are** **2**^{(}^{n}^{2}^{−}^{n}^{)}^{/}^{2}**reflexive and symmetric binary **
** relations on ****A. **

**4. There are 2***^{n}*×

**3**

^{(}

^{n}

^{2}^{−}

^{n}^{)}

^{/}

^{2}**antisymmetric binary relations**

**on**

**A.****Each (i, i) can be either included or excluded. **

**For each pair of (i, j) and (j, i), there are three choices: **

**(a) include (i, j) and exclude (j, i); **

**(b) exclude (i, j) and include (j, i); **

**(c) exclude (i, j) and (j, i). **

**⇒ 2***^{n}*×

**3**

^{(}

^{n}

^{2}^{−}

^{n}^{)}

^{/}

^{2}**.**

**5. There is no general formula for counting the number of **
**transitive binary relations on A. **

**A poset A is called a lattice, if every two elements of A have ****their least upper bound and greatest lower bound in A. **

**Ex. The poset A in the above example is not a lattice. **

**Ex. Let A be the power set of {1, 2, 3}.**^{ }

* Define R on A as follows: a R b iff a*⊂

**b.****The least upper bound (greatest lower bound) of **
**a and b is**^{ }* a*∪

**b**

^{ }*∩*

**(a**

**b).****⇒ A is a lattice. **

**Ex. Define R on the set Z of integers as follows: **

**a R b**** ****iff**** ****4**^{ }**divides**^{ }* a*−

**b.****R is an equivalence relation, and its equivalence classes, ****denoted by [0], [1], [2] and [3], are as follows: **

** [0]={…, −8, −4, 0, 4, 8, …}=*** {4k| k*∈

**Z};**** [1]={…, −7, −3, 1, 5, 9, …}=****{4k****+1*** | k*∈

**Z};**** [2]={…, −6, −2, 2, 6, 10, …}=****{4k****+2*** | k*∈

**Z};**** [3]={…, −5, −1, 3, 7, 11, …}=****{4k****+3*** | k*∈

**Z}.****Ex. Define R on the set Z of integers as follows: **

**a R b**** ****iff**** ****a**^{2}**=****b**^{2}**. **

**R is an equivalence relation, and its equivalence classes ****are {0}, {−1, 1}, {−2, 2}, …, {−****i, i}, … . **

**Ex. Suppose that R is an equivalence relation on {1, 2, …, ****7}, and the induced partition is {{1, 2}, {3}, {4, 5, 7}, **
** {6}}. **

** Then,**^{ }* R = {(1, 1), (2, 2), (1, 2), (2, 1)}*∪

**{(3, 3)}**∪

_{ }**{(4, 4), (5, 5), (7, 7), (4, 5), (5, 4), (4, 7), **
**(7, 4), (5, 7), (7, 5)}**∪**{(6, 6)}. **

**There is a one-to-one correspondence between the set **
**of equivalence relations on {1, 2, …, n} and the set of ****partitions of {1, 2, …, n}. **

**Ex. What is the number of equivalence relations on A****={1, **
** 2, ****..., 6}? **

** Let ****f(m) be the number of onto functions from A to B**^{B}**=**_{ }**{b**_{1}**, b**_{2}**, …, b**_{m}**}, which can be evaluated by the principle **
**of inclusion and exclusion. **

**The answer is equal to **

**f(1)****+****f(2)/2!****+****f(3)/3!****+****f(4)/4!****+****f(5)/5!****+****f(6)/6! **

** = 1+31+90+65+15+1 **

** = 203. **

**Boolean**

^{ }**Algebra **

( , , , ) = + + + + +

*f w x y z* *wx yz* *wxyz* *wx y z* *wx yz*
*wxyz* *wx y z*

### ( ) ( )

### ( ) ( )

## ( ( ) )

( , , , ) = + +

= + + + = +

= 1 + (1 )

*f w x y z* *wxz y* *y* *wx y z* *z* *wxyz*
*wx y z*

*wxz* *wx y* *wxyz* *wx y z*
*wx z* *y z* *wx y* *yz*

*wx z* *y* *y z* *wx y* *z* *y*

+ + +

+ +

+ +

### (

+ +### )

## ( ( ) ) ( ^{(} ^{)} )

= +

*z*
*wx z* + *y z* + *z* *wx y* + *z y*+ *y*

### ( ) ( )

= +

= + + +
*wx z* *y* *wx y* *z*
*wxz* *wx y* *wx y* *wxz*

+ +

**The elements of a finite Boolean algebra can be partially **
**ordered. **

**Suppose that (K, **

### ⋅

**, +) is a Boolean algebra.**

**For**^{ }* a, b*∈

**K,**

**define**

*p*

**a**

**b**

**iff**

**a**### ⋅

^{b}

^{=}

^{a. }**Then, ** _{p}** is a partial ordering. **

** reflexisive****: a**

### ⋅

^{a}

^{=}

^{a }^{⇒}

*p*

^{a}

**a.**** antisymmetric*** : a*p

**b,**

*p*

**b**

**a**

**⇒ a**### ⋅

^{b}

^{=}

^{a,}

^{ }

^{b}### ⋅

^{a}

^{=}

^{b }**⇒ a****=****b. **

** transitive*** : a*p

**b,**

*p*

**b**

**c**

**⇒ a**### ⋅

^{b}

^{=}

^{a,}

^{ }

^{b}### ⋅

^{c}

^{=}

^{b }**⇒ a****=****a**

### ⋅

^{b}

^{=}

^{a}### ⋅

^{(b}### ⋅

^{c)}

^{=}

^{(a}### ⋅

^{b)}### ⋅

^{c }**= a**

### ⋅

^{c }* ⇒ a*p

**c.****The Hasse diagrams for the Boolean algebra of page 12 is **
**depicted below. **

**Notice that**^{ }**0**_{p} **a,**^{ }**a**_{p} **1**** ****for every**^{ }* a*∈

**K.****(0=1**^{ }**and**^{ }**1=30**** ****for the example above) **

* A nonzero element a*∈

**K is called an atom of K,****if**

**b**_{p}

**a**

**implies**

**b****=0**

**or**

**b****=**

**a,**

^{ }*∈*

**where b**

**K.****The Boolean algebra of the example above has three atoms: **

**2, 3, 5. **

**Fact 1. If a is an atom of K,**^{ }**then**^{ }**a**

### ⋅

^{b}

^{=}^{0}

^{ }

^{or}

^{ }

^{a}### ⋅

^{b}

^{=}

^{a}

^{ }

^{for }** every**^{ }* b*∈

**K.****Fact 2. If**^{ }**a**** _{1 }**≠

**a**

_{2}

^{ }

**are two atoms of K,**

^{ }**then**

^{ }

**a**

_{1 }### ⋅

^{a}**2**

**=0.**

^{ }**Fact 3. Suppose that**^{ }**a**_{1}**, a**_{2}**, …, a**_{n}^{ }**are atoms of K, and b is a ****nonzero element in K.**^{ }**Without loss of generality, **
** assume**^{ }**b**

### ⋅

^{a}*≠*

**i****0**

^{ }**for**

^{ }**1**≤

*≤*

**i**

**k,**

^{ }**and**

^{ }

**b**### ⋅

^{a}

**i****=0**

^{ }**else.**

** Then,**** ****b****=****a**_{1 }**+****a**_{2 }**+ … +****a**_{k}**. **

**Fact 4. If K has n atoms,**^{ }**then**^{ }**|K|****=2**^{n}**. **

**For the example above, we have**^{ }**10=2+5,**^{ }**30=2+3+5,**
**and**^{ }**|K|****=2**^{3}**=8. **

**The proofs of the four facts can be found in pages 738 and **
**739 of Grimaldi’s book. **

**Rings **

**Ex. Let R****=****{a, b, c, d, e}, and define + and **

### ⋅

**as follows.**

**(R, +, **

### ⋅

**) is a commutative ring with unity, but without**

**proper divisors of zero.**

** zero:**^{ }**a. **

** unity:**^{ }**b. **

** units:**^{ }**b, c, d, e. **

**Every nonzero element has a multiplicative inverse. **

**Ex. Let R****=****{s, t, v, w, x, y}, and define + and **

### ⋅

**as follows.**

**(R, +, **

### ⋅

**) is a commutative ring with unity.**

** zero:**^{ }**s. **

** unity:**^{ }**t. **

** units:**^{ }**t, y. **

**(R, +, **

### ⋅

**) is not an integral domain,**

**because**

**v**### ⋅

^{w}

^{=}

^{s. }**(R, +, **

### ⋅

**) is not a field,**

**because**

**v, w and x have no****multiplicative inverses.**

**Also notice that**** ****v**

### ⋅

^{v}

^{=}

^{v}### ⋅

^{y}

^{=}

^{x,}

^{ }

^{i.e.,}

^{ }**the cancellation law**

**of multiplication does not hold for this example.**

** **

**Proof of Theorem 14.10 in Grimaldi’s book. **

* • z*∈

**S.**** Let**^{ }**a****=****b**** ^{ }**⇒

^{ }

**b****+(−**

**b)****=**

**z ∈ S.****• For each**^{ }* b*∈

**S,****−**

^{ }*∈*

**b**

**S.**** Let**^{ }**a****=****z**** ^{ }**⇒

^{ }

**z****+(−**

**b)****= −**

**b ∈ S.****• a****+*** b*∈

**S**

^{ }**for all**

^{ }*∈*

**a, b**

**S.*** a, −b*∈

**S****⇒**

^{ }

^{ }

**a****+(−(−**

**b))****= a****+**

**b ∈ S.****When S is finite,**^{ }**we assume**^{ }**S****=****{s**_{1}**, s**_{2}**, …, s**_{n}**}. **

**For every**^{ }* a*∈

**S,**

^{ }

**{a****+**

**s**

_{1}

**, a****+**

**s**

_{2}

**, …, a****+**

**s**

_{n}**}=**

**S.**⇒^{ }**a****=****a****+****s**_{k }**=****s**_{k }**+****a**^{ }**for some 1**≤* k*≤

**n**⇒^{ }**s**_{k }**=*** z*∈

**S**⇒^{ }**z****=****a****+****s**_{l }**=****s**_{l }**+****a**^{ }**for some 1**≤* l*≤

**n**⇒^{ }**s**_{l }**= −*** a*∈

**S.****For 1**≤* a*<

**n,**** gcd(a, n)****=1 ⇒ [a]**^{−1}** exists**^{ }**(i.e.,**** ****[a] is a unit **** of Z**_{n}**); **

** gcd(a, n)****>1 ⇒ [a] is a proper zero divisor of **
** ** **Z**_{n}**. **

**Ex. Find [25]**^{−1}** in Z**_{72}^{ }**(gcd(25, 72)=1). **

** (−23)**×**25+8**×**72=1. **

** (−23)**×**25 ≡ 1 (mod 72). **

** [25]**^{−1}**=[−23]=[49]. **

**Ex. Find x so that 25x ≡ 3 (mod 72). **

** (−23)**×**25 ≡ 1 (mod 72). **

** 3**×**(−23)**×**25 ≡ 3 (mod 72). **

** (−69)**×**25 ≡ 3 (mod 72). **

* x*∈

**[−69]=[3].**

**Ex. gcd(8, 18)=gcd(2**×**4, 2**×**9)=2. **

**⇒ [8]**

### ⋅

^{[9]}^{=}^{[0] }**⇒ [8] is a proper zero divisor of Z****. **

**Ex. How many units and how many proper zero **
** divisors are there in Z**_{72}**? **

** The number of units in Z**_{72}** is equal to the number **
* of integers a such that 1*≤

*<*

**a**

**72 and gcd(a, 72)****=1.**

**The latter can be computed as **
φ**(72)=**φ**(2**** ^{3}**×

**3**

^{2}**)=72**×

**(1**−

^{1}**2)**×**(1**− ^{1}

**3)=24, **
** where **φ**(n) is the Euler’s phi function**^{ }**(refer to **

**Example 8.8 in page 394 of Grimaldi’s book). **

** The number of proper zero divisors in Z**_{72}** is **
** equal to**^{ }**71**−**24=47. **

**Ex. How many units and how many proper zero **
**divisors are there in Z**_{117}**? **

** The number of units in Z**_{117}** is equal to **
** ** φ**(117)=**φ**(3**** ^{2}**×

**13)=117**×

**(1**−

^{1}**3)**×**(1**− ^{1}

**13)=72. **

** The number of proper zero divisors in Z**_{117}** is **
** equal to**^{ }**116**−**72=44. **

**The Chinese Remainder Theorem **

**m**_{1}**, m**_{2}**, …, m**_{k}** : positive integers that are greater than 1 **
* and are prime to one another , where k*≥

**2.**

**0**≤**a***_{i }*≤

**m**

_{i}

^{ }**for 1**≤

*≤*

**i**

**k.****M**_{i }**=****m**_{1}**…m**_{i−1}**m**_{i+1}**…m**_{k}^{ }**for 1**≤* i*≤

**k.****M**_{i}**x***_{i }*≡

**1**

**(mod m**

_{i}**)**

^{ }**for 1**≤

*≤*

**i**

**k.****Then,**^{ }**x****=****a**_{1}**M**_{1}**x**_{1 }**+****a**_{2}**M**_{2}**x**_{2 }**+ … +****a**_{k}**M**_{k}**x**_{k}^{ }**is a solution to **
* x*≡

**a**

_{i }

**(mod m**

_{i}**)**

^{ }**for 1**≤

*≤*

**i**

**k.****Moreover,**^{ }**if**^{ }**y is another solution,**^{ }**then **
* y*≡

**x****(mod m**

_{1}

**m**

_{2}

**…m**

_{k}**).**

**The proof can be found in page 702 of Grimaldi’s book. **

* Ex. x*≡

**14(mod 31),**

^{ }*≡*

**x****16(mod 32),**

^{ }**and**

^{ }*≡*

**x****18(mod 33).**

**(a**_{1}**, a**_{2}**, a**_{3}**)=(14, 16, 18);**^{ }**(m**_{1}**, m**_{2}**, m**_{3}**)=(31, 32, 33);**^{ }

**(M**_{1}**, M**_{2}**, M**_{3}**)=(1056, 1023, 992). **

**M**_{1}**x**** _{1 }**≡

**1**

**(mod m**

_{1}**)**

^{ }**(i.e.,**

**gcd(x**

_{1}

**, m**

_{1}**)=1)**

⇒^{ }**[x**_{1}**]=****[M**_{1}**]**^{−1}**=[1056]**^{−1}**=[2]**^{−1}**=[16]**^{ }**in**

**m****1**

**Z****=****Z**_{31}**. **
**Similarly,**^{ }**[x**_{2}**]=[31]**^{ }**in**

**m****2**

**Z****=****Z**_{32}^{ }**and **
**[x**_{3}**]=[17]**^{ }**in**

**m****3**

**Z****=****Z**_{33}**. **

** Therefore,**^{ }**x****=** **(a**_{1}**M**_{1}**x**_{1 }**+****a**_{2}**M**_{2}**x**_{2 }**+****a**_{3}**M**_{3}**x**_{3}**) mod 32736**_{ }

**= 32688 **
**is a solution. **

**The general solution is **

* y*≡

**x****(mod 32736).**

**In the study of cryptology, we often need to compute **
**b**^{e}** mod n,**^{ }**where b, e and n are large integers. **

**Ex. Compute 5**^{143}** mod 222. **

** Let ****E(i)****=5**^{i}* mod 222, where i*≥

**0 integer.**

⇒^{ }**E(2i)****=****(E(i))**^{2}** mod 222. **

** 143=2**^{7}**+2**^{3}**+2**^{2}**+2**^{1}**+2**^{0}**. **

⇒^{ }**E(143)****=****(E(2**^{7}**)**×**E(2**^{3}**)**×**E(2**^{2}**)**×**E(2**^{1}**)**×**E(2**^{0}**)) mod 222. **

**E(2**^{0}**)=5. **

**E(2**^{1}**)=****(E(2**^{0}**))**^{2}** mod 222=25. **

**E(2**^{2}**)=****(E(2**^{1}**))**^{2}** mod 222=181. **

**E(2**^{3}**)=****(E(2**^{2}**))**^{2}** mod 222=127. **

**E(2**^{4}**)=****(E(2**^{3}**))**^{2}** mod 222=145. **

**E(2**^{5}**)=****(E(2**^{4}**))**^{2}** mod 222=157. **

**E(2**^{6}**)=****(E(2**^{5}**))**^{2}** mod 222=7. **

**E(2**^{7}**)=****(E(2**^{6}**))**^{2}** mod 222=49. **

**E(2**^{8}**)=****(E(2**^{7}**))**^{2}** mod 222=181. **

**Therefore,**^{ }**E(143)**_{ }**=**_{ }**(49**×**127**×**181**×**25**×**5) mod 222 **

** = ((49**×**127) mod 222)**×**((181**×**25**×**5) **
**mod 222) mod 222 **

** = 7**×**203 mod 222 **

** = 89. **

** ** **f **

**(R, +, **

### ⋅

**) −−−−−−−−−−−−> (S,**⊕

**,**

^{~}

**)**

**a ****f(a) **

**b ****f(b) **

**a****+****b ****f(a****+****b)****=*** f(a)*⊕

**f(b)**

**a**### ⋅

^{b }

^{f(a}### ⋅

^{b)}

^{=}

^{f(a)}^{~}

^{f(b) }**Performing + (or **

### ⋅

**) in R and then mapping is “equivalent”****to mapping and then performing **⊕** (or **~**) in S. **

* (a) For any s*∈

**S,**

*∈*

**there exist r**

**R**

**with**

**f(r)****=**

**s****(because**

**f is onto).****s****=****f(r)****=****f(ru**_{R}**)=****f(r)f(u**_{R}**)=****sf(u**_{R}**). **

** Similarly,**** ****s****=****f(u**_{R}**)s. **

**⇒ ****f(u**_{R}**) is the unity of S.**

**(b) Suppose b****=****a**^{−1}**. **

**⇒ ****ab****=****ba****=****u**_{R}

**f(a)f(b)****=****f(ab)****=****f(u**_{R}**)=****u**_{S }**(u**_{S}** is the unity of S) **** Similarly,**** ****f(b)f(a)****=****u**_{S}**. **

**⇒ ****f(a)f(b)****=****f(b)f(a)****=****u**_{S}

**⇒ ****(f(a**^{−1}**)=****) f(b)****=****[f(a)]**^{−1}

**Ex. (following the discussion of the Chinese remainder **
** theorem) **

**(Z**_{32736}**, +, **

### ⋅

**) is a ring,**

^{ }**where 32736=31**×

**32**×

**33.**

**(Z**** _{31 }**×

**Z****×**

_{32 }

**Z**

_{33}**,**⊕

**,**~

**) is a ring,**

^{ }**where**

**(x**_{1}**, x**_{2}**, x**_{3}**)**⊕**(y**_{1}**, y**_{2}**, y**_{3}**)=****(x**_{1 }**+****y**_{1}**,**^{ }**x**_{2 }**+****y**_{2}**,**^{ }**x**_{3 }**+****y**_{3}**)**^{ }**and **
**(x**_{1}**, x**_{2}**, x**_{3}**)**~**(y**_{1}**, y**_{2}**, y**_{3}**)=****(x**_{1 }

### ⋅

^{y}**1**

**,**

^{ }

**x**

_{2 }### ⋅

^{y}**2**

**,**

^{ }

**x**

_{3 }### ⋅

^{y}**3**

**).**

** Define**^{ }**f****: (Z**_{32736}**, +, **

### ⋅

^{) → (Z}**31**×

**Z****×**

_{32 }

**Z**

_{33}**,**⊕

**,**~

**)**

^{ }**as follow:**

**f(x)****=****(x**_{1}**, x**_{2}**, x**_{3}**),**^{ }**where**^{ }**x**_{1 }**=****x mod 31,**^{ }**x**_{2 }**=****x mod 32,**^{ }**and **
**x**_{3 }**=****x mod 33. **

**f**^{ }**is an isomorphism from**^{ }**Z**_{32736}^{ }**to**^{ }**Z**** _{31 }**×

**Z****×**

_{32 }

**Z**

_{33}**.**

**(Refer to Example 14.21 in page 700 of Grimaldi’s**

**book.)**

** 18152**

### ⋅

^{18153}

^{ }

^{in Z}**32736**

^{ }**can be computed as follows.**

**18152**

### ⋅

^{18153}

**=**

_{ }

**f**

^{−1}

**f(18152**### ⋅

^{18153) }** ** **= ****f**^{−1}* (f(18152)*~

**f(18153))**

**=**

**f**

^{−1}**((17, 8, 2)**~

**(18, 9, 3))**

** ** **= ****f**^{−1}**(17**×**18 mod 31,**^{ }**8**×**9 mod 32,**^{ }**2**×**3 mod 33) **

** ** **= ****f**^{−1}**(27, 8, 6) **

**(refer to the example of the Chinese **

** ** ** remainder theorem) **

** ** **= 25416 **

** In general,**^{ }**if**^{ }**n****=****n**** _{1 }**×

**n**

_{2 }**× … ×**

**n**

_{k}**,**

^{ }**where**

^{ }

**n**

_{i }**>1**

^{ }**is an**

**integer**

^{ }**and**

^{ }

**gcd(n**

_{i}

**, n**

_{j}**)=1,**

^{ }**then**

^{ }

**(Z**

_{n}**, +,**

### ⋅

^{)}

^{ }

^{and}

^{ }**(** × ×

**n****1**

**Z****Z****n****2** **…**× **, **⊕**, **~**)**^{ }**are isomorphic. **

**n****k**

**Z**

**As a result,**^{ }**computation on large integers in Z**_{n}** can **
**be achieved with (parallel) computation on smaller **
** integers in** × ×

**n****1**

**Z****Z****n****2** **…**× **. **

**n****k**

**Z**

**Groups **

**Ex.**^{ }* G = {*π

_{0}**,**π

_{1}**,**π

_{2}

**, r**

_{1}

**, r**

_{2}

**, r**

_{3}**}, where**

π_{0}** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**3**
**2**
**1**

**3**
**2**

**1** ** ** π_{1}** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**2**
**1**
**3**

**3**
**2**

**1** ** ** π_{2}** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**1**
**3**
**2**

**3**
**2**
**1**

**r**_{1}** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**3**
**1**
**2**

**3**
**2**

**1** ** r**_{2}** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**2**
**3**
**1**

**3**
**2**

**1** ** r**_{3}** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**1**
**2**
**3**

**3**
**2**

**1** **. **

**(G, **

### ⋅

**) is a (nonabelian) group.**

π**1 **

### ⋅

^{r}**1**

**=**⎟

⎠

⎜ ⎞

⎝

⎛

**2**
**1**
**3**

**3**
**2**

**1** ⎟

⎠

⎜ ⎞

⎝

⎛

**3**
**1**
**2**

**3**
**2**

**1** ** = ** ⎟

⎠

⎜ ⎞

⎝

⎛

**1**
**2**
**3**

**3**
**2**

**1** ** = r**_{3}**. **