5 Algebraic structures

5 Algebraic structures
Englische Aufgaben zum Kapitel 5 Algebraische Strukturen
5.1 Decide, whether the set together with the given binary operation forms a group, a ring or a field, or none
of these. Multiple selections may be possible.
a. set of all positive rational numbers together with multiplication
b. set of all integers together with addition and multiplication
c. set of all rational numbers together with addition
d. set of all real numbers together with addition and multiplication
e. set of all positive integers together with multiplication
f. Z 6 together with addition and multiplication
g. Z 11 together with addition and multiplication
[set … Menge; binary operation … binäre Rechenoperation; group … Gruppe; ring … Ring; field … Körper; integer … ganze Zahl;
rational number … rationale Zahl; real number … reelle Zahl]
5.2 Compute the multiplication table for
a. Z 5 , b. Z 6 ,
and use the tables to find the invertible elements of Z 5 and Z 6 , respectively.
[multiplication table … Multiplikationstafel; invertible … invertierbar]
5.3 Let M be a subset of R, M = {a + bÖ5 | a, b * Q}.
a. Work out a formula for addition and multiplication, respectively, of two elements of M.
b. Compute the inverse with respect to addition of 3 + 4 5 .
c. Compute the inverse element with respect to multiplication.
(1) 1 − 2 5 , (2) − 5 5 , (3) 7 , (4) 3 + 5 .
[subset … Teilmenge; field … Körper; real numbers … reelle Zahlen; inverse element … inverses Element;
with respect to … bezüglich]
 a 0 

 | a, b ∈ R . Examine the properties of M under
5.4 Let M be a set of diagonal 2-by-2 matrices, M = 
 0 b 

addition and multiplication and name the inverse and the neutral element. Decide whether M together
with addition, multiplication, or both, is a group, a ring or a field.
5.5 a. Decide whether the number AT20 43000 4736 446 0000 may (theoretically) be an IBAN.
b. The account number of the donation account of the Society for threatened peoples is 0000723 8909, the
bank code number is 60000. Compute the IBAN.
[account number … Kontonummer; donation account … Spendenkonto; bank code number … Bankleitzahl]
5.6 Use the Euclidean algorithm and find integers u and v such that u ⋅ a + v ⋅ b = gcd(a, b ) .
a. a = 114, b = 98
b. a = 294, b = 119
[Euclidean algorithm … euklidischer Algorithmus; gcd (greatest common divisor) … größter gemeinsamer Teiler]
5.7 Use Euclid’s algorithm to find the inverse of a in Z n .
a. a = 47; n = 101
b. a = 15; n = 47
[Euclid’s algorithm … euklidischer Algorithmus]
5.8 Consider the set of all people and check whether the following relations are reflexive and/or transitive.
Which of the relations are equivalence relations? Reason your decision.
a. A is B’s grandmother.
b. A goes to the same school as B.
c. A is older than B.
d. A is a relative of B.
e. A lives in the same country as B.
f. A earns more money than B.
[equivalence relation… Äquivalenzrelation]
5.9 Consider the following relations R on M and decide whether they are equivalence or order relations.
a. M = Z; aRb ⇔ a − b is an even number.
b. M = N; aRb ⇔ a ≤ b
[order relation… Ordnungsrelation; even number … gerade Zahl]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
5 Algebraic structures: solutions
Lösungen zu: Englische Aufgaben zum Kapitel 5 Algebraische Strukturen
5.1 a. group
b. ring
c. group
d. ring, field
e. none of these algebraic structures
f. ring
g. ring, field
5.2 a. multiplication table for Z 5
×5
0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
1
3
3
0
3
1
4
2
4
0
4
3
2
1
invertible elements: 1 [inverse: 1], 2 [inverse: 3], 3 [inverse: 2], 4 [inverse: 4]
5.3 a. multiplication table for Z 6
×6
0
1
2
3
4
5
0
0
0
0
0
0
0
1
0
1
2
3
4
5
2
0
2
4
0
2
4
3
0
3
0
3
0
3
4
0
4
2
0
4
2
5
0
5
4
3
2
1
invertible elements: 1 [inverse: 1], 5 [inverse: 5]
) (
(
(a + b 5 )⋅ (c + d 5 ) = (ac + 5bd) + (ad + bc) 5
)
5.4 a. a + b 5 + c + d 5 = (a + c ) + (b + d) 5 ;
b. − 3 − 4 5
(
)
c. (1) − 191 1 + 2 5 , (2) −
1
5 5
, (3)
1
7
, (4)
1
4
(3 − 5 )
5.5 addition: satisfied properties: associativity and commutativity, inverse Element of A ∈ M : −A , neutral
0 0
 . Thus, M together with addition forms an Abelian group.
Element: 
0 0
 1 0
 . However, for
multiplication: satisfied properties: associativity, commutativity, neutral Element: 
0 1
a = 0 or b = 0 (but not both), the matrix A ∈ M is singular, which means, that it is not invertible. So, not
every element of M has a multiplicative inverse. Thus, M together with multiplication is not a group.
Therefore, M is neither a ring, nor a field.
[Abelian group … Abel’sche Gruppe (kommutative Gruppe);
5.6 a. u = -6, v = 7
b. u = -2, v = 5
5.7 Since 101 and 47 are prime numbers, both tasks are solvable.
a. the inverse of 47 in Z 101 is 43.
b. the inverse of 15 in Z 47 is 22.
[prime numbers … Primzahlen; solvable … lösbar]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
5 Algebraic structures: solutions
Lösungen zu: Englische Aufgaben zum Kapitel 5 Algebraische Strukturen
5.8 a. neither reflexive, nor transitive. Therefore, the relation is not an equivalence relation (additionally,
the relation is not symmetric).
b. reflexive, transitive. The relation is an equivalence relation, since the relation is also symmetric.
c. transitive but not reflexive. Thus, it is not an equivalence relation (additionally, the relation is not
symmetric.)
d. reflexive, transitive. The relation is an equivalence relation, since it is also symmetric.
e. reflexive, transitive. The relation is an equivalence relation, since it is also symmetric.
f. not reflexive, but transitive. Thus, the relation is not an equivalence relation.
5.9 a. equivalence relation: reflexive: a − a = 0 is even, thus aRa; symmetric: b − a = −(a − b ) , and so, if a – b
is even, b – a is even as well; transitive: if a – b is even and b – c is even, so is a – c.
b. order relation: antisymmetric: if a ≤ b and b ≤ a , then a = b; transitive: if a ≤ b and b ≤ c , then
a≤c.
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner