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Selected Topics in Mathematics
2
8
¹
1˜Ù
Mappings and Operations
1 Ù
Introduction to Groups
13
1nÙ
Introduction to linear algebra: matrix theory
15
1oÙ
Complex Numbers
19
1ÊÙ
Polynomials and Complex Polynomials
21
5
3
4
8 ¹
1˜Ù
Mappings and Operations
1. Let S = {x, y, z, w} and T = {1, 2, 3, 4}, and define α : S → T and
β : S → T by α(w) = 2, α(x) = 4, α(y) = 1, α(z) = 2 and β(w) = 4, β(x) =
2, β(y) = 3, β(z) = 1.
(a) Is α one-to-one? Is β one-to-one? Is α onto? Is β onto?
(b) Let A = {w, y} and B = {x, y, z}. Determine each of the following subsets
of T : α(A), β(B), α(A ∩ B), β(A ∪ B).
2. Let α, β, γ be mappings from Z to Z defined by α(n) = 2n, β(n) = n + 1,
and γ(n) = n3 for each n ∈ Z.
(a) Which of the three mappings are onto?
(b) Which of the three mappings are one-to-one?
(c) Determine α(N), β(N), and γ(N).
3. Assume S and T as in Example 1.1.
(a) How many mappings are there from S to T ?
(b) How many mappings are there from S onto T ?
(c) How many one-to-one mappings are there from S to T ?
4. Assume that S and T are sets and α : S → T and β : S → T. Complete
each of the following statements.
(i) α is not onto iff for some y ∈ T · · ·
(ii) α is not one-to-one iff · · ·
(iii) α 6= β iff · · ·
5
1˜Ù Mappings and Operations
6
5. Assume that S and T are finite sets containing m and n elements, respectively.
1. How many mappings are there from S to T ?
2. How many one-to-one mappings are there from S to T ?
6. A mapping f : R → R is onto iff each horizontal line intersects the graph
of f at least once.
(a) Formulate a similar condition for f : R → R to be one-to-one
(b) Formulate a similar condition for f : R → R to be both one-to-one and onto.
7. Each f below defines a mapping from R to R. Determine which of three
mappings are onto and which are one-to-one.
(a) f (x) = 2x.
(b) f (x) = x − 4.
(c) f (x) = x3 .
(d) f (x) = x2 + x.
(e) f (x) = ex .
(f) f (x) = tan x.
8. For each ordered pair (a, b) of integers define a mapping αa,b : Z → Z by
αa,b (n) = an + b.
(a) For which pairs (a, b) is αa,b onto?
(b) For which pairs (a, b) is αa,b one-to-one?
9. Let α, β, γ be mappings from Z to Z defined by α(n) = 2n, β(n) = n + 1
and γ(n) = n2 . Write a formula for each of the following compositions. Also
determine the image in each case.
7
(a) α ◦ α
(b) β ◦ α
(c) γ ◦ α
(d) α ◦ β
(e) β ◦ β
(f) γ ◦ β
(g) α ◦ γ
(h) β ◦ γ
(i) γ ◦ γ.
10. Prove that if α : S → T , then α ◦ ιS = α and ιS ◦ α = α.
11. Describe the inverse of the mapping β in Example 9.
12. Each f below defines an invertible mapping from R to R. Write a formula
for the inverse in each case.
(a) f (x) = 2x.
(b) f (x) = x − 4.
(c) f (x) = 2x − 4.
(d) f (x) = x3 .
13. Prove that the inverse of an invertible mapping is invertible.
14. If α : S → T , and A and B are subsets of S. Prove that
(i) α(A ∪ B) = α(A) ∪ α(B).
(ii) α(A ∩ B) ⊆ α(A) ∩ α(B); Give an example (specific S, T, A, B, and α) to
show that equality need not hold.
1˜Ù Mappings and Operations
8
15. Prove that a mapping α : S → T is one-to-one iff
α(A ∩ B) = α(A) ∩ α(B)
for every pair of subsets A and B of S.
16. Give an example of sets S, T, and U and mappings α : S → T and
β : T → U such that β ◦ α is onto but α is not onto.
17. Give an example of sets S, T, and U and mappings α : S → T and
β : T → U such that β ◦ α is one-to-one but β is not one-to-one.
18. Prove that if α : S → T, β : T → U, γ : T → U, α is onto, and
β ◦ α = γ ◦ α, then β = γ.
19. Prove that if β : S → T, γ : S → T, α : T → U, α is one-to-one, and
α ◦ β = α ◦ γ, then β = γ.
20. Give an example to show that the condition “α is onto” cannot be
omitted from Problem 18.
21. Give an example to show that the condition “α is one-to-one” cannot be
omitted from Problem 19.
22. Complete the following statement: A mapping α : S → T is not invertible iff α is not one-to-one · · ·
23. Let α : S → T and β : T → U. Prove each of the following statements.
(a) If α and β are invertible, then β ◦ α is invertible.
(b) If β ◦ α is invertible, then β is onto and α is one-to-one.
24. What of the following define operations on the set of integers? Of those
that do, which are associative? Which are commutative? Which have identity
element?
(a) m ∗ n = mn + 1
(b) m ∗ n = (m + n)/2
(c) m ∗ n = m
9
(d) m ∗ n = mn2
(e) m ∗ n = m2 + n2
(f) m ∗ n = 2mn
(g) m ∗ n = 3
(h) m ∗ n =
√
mn
25. Verify that the operation in Example 1 has no identity element.
26. Does (m, n) → mn define an operation on the set of all integers?
27. If ∗ is an operation on S, T is a subset of S, and T is closed with respect
to ∗, then two of the following three statements are necessarily true but one may
be false. Which two are true?
(a) If ∗ is associative on S, then ∗ is associative on T.
(b) If there is an identity element for ∗ on S, then there is an identity element
for ∗ on T.
(c) If ∗ is commutative on S, then ∗ is commutative on T.
28. Assume that ∗ is an operation on S. Complete each of the following
statements.
(a) ∗ is not associative iff a ∗ (b ∗ c) 6= (a ∗ b) ∗ c · · ·
(b) ∗ is not commutative iff a ∗ b 6= b ∗ a.
(c) e ∈ S is not an identity element for ∗ iff · · ·
(d) There is no identity element for ∗ iff for each e ∈ S there is an element a ∈ S
such that · · ·
29.
(a) Complete the following Cayley table in such a way that u becomes an identity
element. In How many ways can this be done?
1˜Ù Mappings and Operations
10
(b) Can the table be completed in such a way that u and v both become identity
elements? Why or why not?
(c) Prove: An operation ∗ on a set S can have at most one identity element.
30.
(a) Determine the smallest subset A of Z such that 2 ∈ A and A is closed with
respect to addition.
(b) Determine the smallest subset B of Q such that 2 ∈ B and B is closed with
respect to addition and division.
31. With S = {a, b}, the set M (S) contains four elements; denote these by
π, ρ, σ, τ , defined as follows:
π(a) = a ρ(a) = a σ(a) = b τ (a) = b
π(b) = a ρ(b) = b σ(b) = a τ (b) = b.
(a) Construct the Cayley table for composition ◦ as an operation on M (S) =
{π, ρ, σ, τ }.
(b) What is the identity element?
(c) Is ◦ commutative as an operation on M (S).
(d) Which elements of M (S) are invertible?
(e) Is ◦ commutative as an operation on the set of invertible elements in M (S)?
32. Verify that if S contains more than one element, then composition is not
a commutative operation on M (S).
33. Consider the operation ◦ on the set A in Example 4.2.
(a) Verify that α1,0 is an identity element.
(b) Prove that each αa,b ∈ A is invertible by verifying that it is one-to-one and
onto.
11
(c) Determine (c, d) so that αc,d is an inverse of αa,b .
34. Give a geometric interpretation of αa,b in Example 4.2 under each of the
following conditions?
(a) 0 < a < 1 and b = 0.
(b) a < 0 and b = 0.
(c) c = 1 and b < 0.
35. Let B and C denote the following subsets of A in Example 4.2:
B = {αa,0 |a ∈ R and a 6= 0}
(1.1)
C = {α1,b |b ∈ R}.
(1.2)
(a) Verify that ◦ is an operation on B. Is it associative? Commutative? Is there
an identity element?
(b) Verify that ◦ is an operation on C. Is it associative? Commutative? Is there
an identity element?
(c) Verify that each mapping in A is the composition of a mapping in B and a
mapping in C.
36. Consider Example 4.2 and let D denote the smallest subset of A such
that α1,1 ∈ D and D is closed with respect to ◦. Determine D.
37. Prove that if α : S → T, β : T → U, and γ : U → V are any mappings,
then γ ◦ (β ◦ α) = (γ ◦ β) ◦ α.
38. Is composition an operation on the set of all continuous mappings from
R to R?
39. Prove that if V is a vector space, then composition is an operation on
the set of all linear transformations from V to V.
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1˜Ù Mappings and Operations
1 Ù
Introduction to Groups
1. Determine which of the following sets of numbers from groups with respect
to the given operations. For those that do, give the identity element and the
inverse of each element. For those that do not, give a reason.
(a) {1}, multiplication.
(b) {0}, multiplicaiton.
(c) All nonzero rational numbers, multiplication.
(d) All rational numbers, addition.
(e) All rational numbers, multiplication.
(f) {−1, 1}, multiplication.
(g) {−1, 0, 1}, addition.
(h) All integers, multiplication.
(i) {n|n = 10k for some k ∈ Z}, addition.
(j) All nonzero rational numbers, division.
(k) All integers, subtraction.
2. (a) Verify that {2m |m ∈ Z} is a group with respective to multiplication.
(b) Verify that {2m 3n |m, n ∈ Z} is a group with respect to multiplication.
3. Verify that M (2, Z), the set of all 2 × 2 matrices with integers as entries,
forms a group with respect to matrix addition.
4. If |S| > 1, then M (S) is not a group with respective to composition.
Why?
5. Let F denote the set of all functions f : R → R. For f, g ∈ F, define f + g
by (f + g)(x) = f (x) + g(x) for all x ∈ R. Then f + g ∈ F. Verify that with this
operation F is a group.
13
1 Ù Introduction to Groups
14
6. Let H = {f |f : R → R and f (x) 6= 0 for all x ∈ R}. For f, g ∈ H, define
f g by (f g)(x) = f (x)g(x) for all x ∈ R. Then f g ∈ H. Verify that with this
operation H is a group. How does this group differ from the group of invertible
mappings in M (R)?
7. Verify that the set of all invertible 2 × 2 matrices with real numbers as
entries forms a group with respect to matrix multiplication.
8. Verify the associative law for the operation ∗ in Example 5.5.
9. For example 5.8, verify the claim that the inverse of αa,b is αa−1 ,−a−1 b .
10. If {a, b} with operation ∗ is to be a group, with a the identity element,
then what must the Cayley table be?
11. If {x, y, z} with operation ∗ is to be a group, with x the identity element,
then what must the Cayley table be?
12. Prove that if G is a group, a ∈ G, and a ∗ b = b for some b ∈ G, then a
must be the identity element of G.
13. Formulate a question like that in Problem 5.13 with Theorem 5.1(b) in
place of Theorem 5.1(a). Answer it.
1nÙ
Introduction to linear algebra: matrix
theory
3
0
, B =
1. Consider the matrices A =
−1
2
1 1
1 5 2
6 1 3
D=
−1 0 1 , E = −1 1 2 .
3 2 4
4 1 3
"
4 −1
0
2
#
"
,C =
1 4 2
3 1 5
#
,
2. Using the matrices in above exercise, in each part compute the given
expression (where possible)
(i) (D − E)T (ii) B − B T (iii) (CD)E (iv) tr(BC)
3. Find matrices A, x, and b that express the given system of linear equation
as a single matrix equation Ax = b and write out this matrix equation
x1 − 2x2 + 3x3 = −3
2x1 + x2 = 0
− 3x2 + 4x3 = 1
x1 + x3 = 5
4. Find all values
1 2
[2 2 k]
2 0
0 3
of k, if any, that satisfy the equation.
0
2
3
2 = 0.
1
k
5. Let I be the n × n matrix whose entry in row i and column j is
(
1, i = j;
0, i 6= j.
Show that AI = IA = A for every n × n matrix A.
6. True or False Exercises.
15
1nÙ Introduction to linear algebra: matrix theory
16
1 2 0
(a) The matrix 2 0 3
has no main diagonal.
0 3 1
(b) An m × n matrix has m column vectors and n row vectors.
(c) If A and B are 2 × 2 matrices, then AB = BA
(d) The ith row vector of a matrix product AB can be computed by mutliplying A by the ith row vector of B.
(e) Fir every matrix A, it is true that (AT )T = A.
(f) If A and B are square matrices of the same order, then tr(AB) =
tr(A)tr(B).
(g) If A and B are square matrices of the same order, then (AB)T = AT B T .
(h) For every square matrix A, it is true that tr(AT ) = tr(A).
(i) If A is a 6 × 4 matrix and B is an m × n matrix such that B T AT is a
2 × 6 matrix, then m = 4 and n = 2.
(j) If A is an n × n matrix and c is a scalar, then tr(cA) = ctr(A).
(k) If A, B and C are matrices of the same size such that A − C = B − C,
then A = B.
(l) If A, B and C are square matrices of the same order such that AC = BC,
then A = B.
(m) If AB + BA is defined, then A and B are square matrices of the same
order.
(n) If B has a column of zeros, then so does AB if this product is defined.
(o) If B has a column of zeros, then so does BA if this product is defined.
"
#
3 1
7. Let A =
, compute the inverse of A.
5 2
"
#
6
4
8. Let A =
, compute the inverse of A.
−2 −1
"
#
cos α sin α
9. Let A =
, compute the inverse of A.
− sin α cos α
17
"
10. If A−1 =
2 −1
3
#
, fine A.
5
"
11. Let A be the matrix
2 0
4 1
#
. Compute A3 and P (A), where p(x) = x−2
or p(x) = 2x2 − x + 1.
12. Let A be the diagonal matrix with the product of all entries on the main
diagonal being not zero. Show that A is invertible and find its inverse.
13. Determinewhether the following matrix is invertible. If so, find the
1 1 1
inverse.
1
0
0
0 1 1
14. Solve the linear system by inverting the coefficient matrix.
− x1 + 5x2 = 4
− x1 − 3x2 = 1
15. True or False Exercises.
(a) Two n × n matrices, A and B, are inverses of one another if and only if
AB = BA = 0.
(b) For all square matrices A and B of the same size, it is true that (A+B)2 =
A2 + 2AB + B 2 .
(c) For all square matrices A and B of the same size, it is true that A2 −B 2 =
(A − B)(A + B).
(d) If A and B are invertible matrices of the same size, then AB is invertible
and (AB)−1 = A−1 B −1 .
(e) If A and B are matrices such that AB is defined, then it is true that
(AB)T = AT B T .
"
(f) The matrix A =
a b
c d
#
is invertible if and only if ad − bc 6= 0.
(g) If A and B are matrices of the same size and k is a constant, then
(kA + B)T = kAT + B T .
(h) If A is an invertible matrix, then so is AT .
1nÙ Introduction to linear algebra: matrix theory
18
(i) If p(x) = a0 + a1 x + a2 x2 + · · · + am xm and I is an identity matrix, then
p(I) = a0 + a1 + · · · + am .
(j) A square matrix containing a row or column of zeros cannot be invertible.
(k) The sum of two invertible matrices of the same size must be invertible.
1oÙ
Complex Numbers
√
1. Verify that Q( 2) is a field.
2. Solve the equation 3x + 7 = 6 in Z13 .
3. Prove that Zn is a field if and only if n is a prime.
√
4. If d ∈ Q and d is not a perfect square show that Q( d) is a subfield of R.
5. Let w3 = 1, w 6= 1. Show then that 1 + w + w2 = 0.
6. Let w3 = 1, w 6= 1, so that from Exercise 5, 1 + w + w2 = 0, or w2 =
−1 − w. Let Q(w) = {a + bw : a, b ∈ Q}. Define arithmetic on Q(w) by algebraic
manipulation. Show that Q(w) is a subfield of C.
7. Show that there are infinitely many transcendentals.
(i) For n ∈ N let Pn = {all ractional polynomials of degree ≤ n}. Show that Pn
is countable.
(ii) For n ∈ N let Qn = {all roots for polynomials in Pn }. Show that Qn is countable.
(iii) Let A be the set of algebraic numbers over Q. Show that A is countable.
(iv) Let T be the set of transcendental numbers over Q. Show that T is uncountable.
8. Prove that the LUB property for R is equivalent to the nested intervals
property.
9. Let z = 4 + 7i, w = 6 − i. Find
(i) z¯, w,
¯ |z|, |w|.
(ii) z + w, zw, z/w.
(iii)
√
z
(iv) z in polar form
19
1oÙ Complex Numbers
20
(v) z 5
(vi) Solve the equation for Z : zZ + 6w = 1 − i.
10.
Let z = 1 +
√
3i. Find:
(i) z 26 .
(ii) The five distinct fifth roots of z.
1ÊÙ
Polynomials and Complex Polynomials
1. Prove that if P (x), Q(x) ∈ F [x], then
deg P (x)Q(x) = deg P (x) + deg Q(x)
and
deg (P (x) ± Q(x)) ≤ max{deg P (x), deg Q(x)}.
2. Verify the F [x] is a commutative ring by showing that the ring axioms
hold.
3. Let S be a subring of the field F (such as Z in R). Let S[x] consist of the
polynomials in F [x] with coefficients from S. Show that S[x] is a subring of F [x].
Recall that to show a subset is a subring we must only show that it is nonempty
and closed under addition, subtraction and multiplication.
4. The following theorem can be proved:
Theorem 5.1. Given (n+1) distinct values x0 , x1 , . . . , xn is a field F and (n+1)
other values y0 , y1 , . . . , yn in F , then there exists a unique polynomial f (x) ∈ F [x]
of degree ≤ n such that f (xi ) = yi for i = 0, 1, . . . , n.
The proof is from linear algebra. Suppose f (x) = a0 + a1 x + · · · + an xn with
the ai considered as variables. Using f (xi ) = y1 set up the (n + 1) × (n + 1)
system of equations:
a0 + a1 x0 + · · · + an xn0 = y0
(5.1)
a0 + a1 x1 + · · · + an xn1 = y1
(5.2)
···
(5.3)
···
(5.4)
a0 + a1 xn + · · · + an xnn = yn .
(5.5)
The matrix of this system (with the ai as unknowns) is called the Vandermonde
matrix, and if the xi are distinct it can be shown to have a nonzero determinant
and thus there exists a unique solution for a0 , a1 , . . . , an .
21
1ÊÙ Polynomials and Complex Polynomials
22
(a) Find the unique polynomial P (x) ∈ R[x] of deg ≤ 2 that satisfies P (0) =
1, P (1) = 2, P (2) = 2.
(b) Find the unique polynomial P (x) ∈ Z7 [x] of deg ≤ 2 that satisfies
P (0) = 1, P (1) = 2, P (2) = 2. Z7 is the field of integers modulo 7.
5. (a) Use the theorem in Exercise 4 to prove that if F is an infinite field
and f (x), g(x) ∈ F [x] with f (s) = g(s) for all s ∈ F, then f (x) and g(x) are the
same polynomial.
(b) Show this is not necessarily true over a finite field.
6. Use the division algorithm to find the quotient and remainder for the
following pairs of polynomials in the indicated polynomial rings. (a) f (x) =
x3 + 5x2 + 6x + 1, g(x) = x − 1 in R[x].
(b) f (x) = x3 + 5x2 + 6x + 1, g(x) = x − 1 in Z5 [x].
(c) f (x) = x3 + 5x2 + 6x + 1, g(x) = x − 1 in Z13 [x].
7. Use the Euclidean algorithm to find the gcd of the following pairs of
polynomials in Q[x].
(a) f (x) = 2x3 − 4x2 + x − 2, g(x) = x3 − x2 − x − 2.
(b) f (x) = x3 + x2 + x + 1, g(x) = x3 − 1.
8. Suppose F is a subfield of K. Then from Exercise 3, F [x] is a subring of
K[x]. Suppose f (x), g(x), h(x) ∈ K[x] and suppose f (x) = g(x)h(x). Prove: If
any two of these are in F [x], then so is the third.
9. Show that f (x) = x2 +x+4 is irreducible in R[x] but completely factorizes
in C[x] and give its factorization. This same polynomial completely factorizes in
Z3 [x]; give its factorization there.
10. Formally carry out the derivation of the quadratic formula and show
that it holds over any F of characterisitic 6= 2-that is, 2 6= 0 in F.
11. Show that if z ∈ C, then (x − z)(x − z¯) ∈ R[x].
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