10. Is S= {(a, b)|a +b= 0} a subring of ZxZ Jastify your answer. I1. Let S be the subset of MR) consisting of all matrices of the form ) (4) Prove that S'is a ring isa right identity in S(meaning that AJ = A for (b) Show that J every A in S). (e) Show that Jis not a left identity in S by finding a matrix Bin S such that JB + B. For more information about S, sec Exercise 41. 12. Let 49 denote the set fa t bla, beZ}. Show that Z[g is a subring of C. 13. Let ZV2]denote the set {a + BV2|4, beZ). Show that Z[V2] is a subring of R. Soe Example 20.] 14. Let Tbe the ring in Example 8. Let S = {S€T\f{2) = 0}. Prove that Sis a subring of T. 15. Write out the addition and multiplication tables for
10. Is S= {(a, b)|a +b= 0} a subring of ZxZ Jastify your answer. I1. Let S be the subset of MR) consisting of all matrices of the form ) (4) Prove that S'is a ring isa right identity in S(meaning that AJ = A for (b) Show that J every A in S). (e) Show that Jis not a left identity in S by finding a matrix Bin S such that JB + B. For more information about S, sec Exercise 41. 12. Let 49 denote the set fa t bla, beZ}. Show that Z[g is a subring of C. 13. Let ZV2]denote the set {a + BV2|4, beZ). Show that Z[V2] is a subring of R. Soe Example 20.] 14. Let Tbe the ring in Example 8. Let S = {S€T\f{2) = 0}. Prove that Sis a subring of T. 15. Write out the addition and multiplication tables for
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#12 on the picture.
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77
10. Is S= {(a, b)|a + b = 0} a subring of Z x Z? Justify your answer.
11. Let S be the subset of M(R) consisting of all matrices of the form
(a) Prove that S is a ring.
(b) Show that J =
every A in S).
(c) Show that Jis not a left identity in S by finding a matrix Bin S such that
is a right identity in S (meaning that AJ = A for
JB + B.
For more information about S, see Exercise 41.
12. Let Z denote the set {a + bi |a, beZ}. Show that Z[] is a subring of C.
13. Let Z[V2] denote the set {a + bV2|a, beZ}. Show that Z[V2] is a subring
of R. See Example 20.]
14. Let Tbe the ring in Example 8. Let S = {feT|f(2) = 0}. Prove that S is a
subring of T.
15. Write out the addition and multiplication tables for
(a) Zz x Z;
(b) Zz × Z,
(c) Z, x Z,
-G )-
16. Let A =
and 0 =
in M(R). Let S be the set of all matrices B
such that AB = 0.
(a) List three matrices in S. [Many correct answers are possible.]
(b) Prove that Ss is a subring of M(R). [Hint: If B and Carc in S, show that
B + Cand BC are in S by computing A(B + C) and A(BC).]
17. Define a new multiplication in Z by the rule: ab = 0 for all a, b, eZZ Show that
with ordinary addition and this new multiplication, Z is a commutative ring.
18. Define a new multiplication in Z by the rule: ab = 1 for all a, b, EZ. With
ordinary addition and this new multiplication, is Z is a ring?
19. Let S = {a, b, c} and let P(S) be the set of all subsets of S; denote the
elements of P(S) as follows:
S = {a, b, c}; D = {a, b}; E= {a, c}; F= {b, c};
A = {a}; B= {b}; C= {c}; 0 = Ø.
Define addition and multiplication in P(S) by these rules:
M + N = (M – N) U (N – M)
and
MN = MON.
Write out the addition and multiplication tables for P(S). Also, see Exercise 44.
R 20 Show that the subset P -
ibring Doer P hare](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F99849ad5-a7fc-4a18-a0b9-78ad11535b08%2Fw4013x_processed.png&w=3840&q=75)
Transcribed Image Text:po Thomas W. Hungerford - Abstrac X
b My Questions | bartleby
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of 621
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77
10. Is S= {(a, b)|a + b = 0} a subring of Z x Z? Justify your answer.
11. Let S be the subset of M(R) consisting of all matrices of the form
(a) Prove that S is a ring.
(b) Show that J =
every A in S).
(c) Show that Jis not a left identity in S by finding a matrix Bin S such that
is a right identity in S (meaning that AJ = A for
JB + B.
For more information about S, see Exercise 41.
12. Let Z denote the set {a + bi |a, beZ}. Show that Z[] is a subring of C.
13. Let Z[V2] denote the set {a + bV2|a, beZ}. Show that Z[V2] is a subring
of R. See Example 20.]
14. Let Tbe the ring in Example 8. Let S = {feT|f(2) = 0}. Prove that S is a
subring of T.
15. Write out the addition and multiplication tables for
(a) Zz x Z;
(b) Zz × Z,
(c) Z, x Z,
-G )-
16. Let A =
and 0 =
in M(R). Let S be the set of all matrices B
such that AB = 0.
(a) List three matrices in S. [Many correct answers are possible.]
(b) Prove that Ss is a subring of M(R). [Hint: If B and Carc in S, show that
B + Cand BC are in S by computing A(B + C) and A(BC).]
17. Define a new multiplication in Z by the rule: ab = 0 for all a, b, eZZ Show that
with ordinary addition and this new multiplication, Z is a commutative ring.
18. Define a new multiplication in Z by the rule: ab = 1 for all a, b, EZ. With
ordinary addition and this new multiplication, is Z is a ring?
19. Let S = {a, b, c} and let P(S) be the set of all subsets of S; denote the
elements of P(S) as follows:
S = {a, b, c}; D = {a, b}; E= {a, c}; F= {b, c};
A = {a}; B= {b}; C= {c}; 0 = Ø.
Define addition and multiplication in P(S) by these rules:
M + N = (M – N) U (N – M)
and
MN = MON.
Write out the addition and multiplication tables for P(S). Also, see Exercise 44.
R 20 Show that the subset P -
ibring Doer P hare
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