Download MATH 3613 Homework Set 1 1. Prove that “not-Q ⇒ not

MATH 3613
Homework Set 1
1. Prove that “not-Q ⇒ not-P ” implies “P ⇒ Q”
2. Prove that if m and n are even integers, then n + m is an even integer.
3. Prove that if n is an odd integer, then n2 is an odd integer.
4. Prove that if n is an integer and n2 is odd, then n is odd.
5. Prove, by the contrapositive method, that if c is an odd integer then the equation n2 + n − c = 0 has no
integer solution.
6. Prove, by mathematical induction, that if n ≥ 5 then 2n > n2 .
7. Prove by mathematical induction that
n
X
n(n + 1)(2n + 1)
i2 =
6
i=1
,
∀ n ∈ Z+
8. Prove the following identities.
(a)
B ∩ (C ∪ D) = (B ∩ C) ∪ (B ∩ D)
(b)
B ∪ (C ∩ D) = (B ∪ C) ∩ (B ∪ D)
(c)
C = (C − A) ∪ (C ∩ A)
9. Describe each set in set-builder notation:
(a) All positive real numbers.
(b) All negative irrational numbers.
(c) All points in the coordinate plane with rational first coordinate.
(d) All negative even integers greater than −50.
10. Which of the following sets are nonempty?
(a) r ∈ Q | r2 = 2
(b) r ∈ R | r2 + 5r − 7 = 0
(c) t ∈ Z | 6t2 − t − 1 = 0
1
.
2
11. Is B is a subset of C when
(a) B = Z and C = Q?
(b) B = all solutions of x2 + 2x − 5 = 0 and C = Z?
(c) B = {a, b, 7, 9, 11, −6} and C = Q?
12. In each part find B − C, B ∩ C, and B ∪ C.
(a) B = Z and C = Q.
(b) B = R and C = Q.
(c) B = {a, b, c, 1, 2, 3, 4, 5, 6} and C = {a, c, e, 2, 4, 6, 8}.
13. Let A, B be subsets of U . Prove De Morgan’s laws:
(a)
U − (A ∩ B) = (U − A) ∪ (U − B)
(b)
U − (A ∪ B) = (U − A) ∩ (U − B)
14.
(a) Give an example of a function that is injective but not surjective.
(b) Give and example of a function that is surjective but not injective.
15. Prove that f : R → R
:
f (x) = 2x + 1
16. Prove that f : R → R
:
f (x) = −3x + 5
is injective.
is surjective.
17. Let B and C be nonempty sets. Prove that the function
f :B×C → C ×B
given by f (x, y) = (y, x) is a bijection.
3
Homework Set 2
(Homework Problems from Chapter 1)
Problems from Section 1.1.
1.1.1. Let n be an integer. Prove that a and c leave the same remainder when divided by n if and only if
a − c = nk for some k ∈ Z.
1.1.2, Let a and b be integers with c 6= 0. Then there exist unique integers q and r such that
(i)
a = cq + r
(ii)
0 ≤ r < |c| .
1.1.3. Prove that the square of any integer a is either of the form 3k or of the form 3k + 1 for some integer
k.
1.1.4. Prove that the cube of any integer has exactly one of the forms 9k, 9k + 1, or 9k + 8.
Problems from Section 1.2
1.2.1.
(a) Prove that if a | b and a | c then a | (b + c).
(b) Prove that if a | b and a | c, then a | (br + ct) for any r, t ∈ Z.
1.2.2. Prove or disprove that if a | (b + c), then a | b or a | c.
1.2.3. Prove that if r ∈ Z is a non-zero solution of x2 + ax + b = 0 (where a, b ∈ Z), then r | b.
1.2.4. Prove that GCD(a, a + b) = d if and only if GCD(a, b) = d.
1.2.5. Prove that if GCD(a, c) = 1 and GCD(b, c) = 1, then GCD(ab, c) = 1.
1.2.6. (a) Prove that if a, b, u, v ∈ Z are such that au + bv = 1, then GCD(a, b) = 1.
(b) Show by example that if au + bv = d > 0, then GCD(a, b) need not be d.
Problems from Section 1.3
1.3.1. Let p be an integer other than 0, ±1. Prove that p is prime if and only if for each a ∈ Z, either
GCD(a, p) = 1 or p | a.
1.3.2
Let p be an integer other than 0 ± 1 with this property: Whenever b and c are integers such that p | bc,
then p | c or p | b. Prove that p is prime.
1.3.3. Prove that if every integer integer n > 1 can be written in one and only one way in the form
n = p1 p2 · · · pr
where the pi are positive primes such that p1 ≤ p2 ≤ · · · ≤ pr .
1.3.4. Prove that if p is prime and p | an , then pn | an .
1.3.5.
(a) Prove that there exist no nonzero integers a, b such that a2 = 2b2 .
4
(b) Prove that
√
2 is irrational.
5
Homework Set 3
(Problems from Chapter 2)
Problems from §2.1
2.1.1. Prove that a ≡ b (mod n) if and only if a and b leave the same remainder when divided by n.
2.1.2. If a ∈ Z, prove that a2 is not congruent to 2 modulo 4 or to 3 modulo 4.
2.1.3. If a, b are integers such that a ≡ b (mod p) for every positive prime p, prove that a = b.
2.1.4. Which of the following congruences have solutions:
(a) x2 ≡ 1 (mod 3)
(b) x2 ≡ 2 (mod 7)
(c) x2 ≡ 3 (mod 11)
2.1.5. If [a] = [b] in Zn , prove that GCD(a, n) = GCD(b, n).
2.1.6. If GCD(a, n) = 1, prove that there is an integer b such that ab = 1 (mod n).
2.1.7. Prove that if p ≥ 5 and p is prime, then either [p]6 = [1]6 or [p]6 = [5]6 .
Problems from §2.2
2.2.1. Write out the addition and multiplication tables for Z4 .
2.2.2. Prove or disprove: If ab = 0 in Zn , then a = 0 or b = 0.
2.2.3. Prove that if p is prime then the only solutions of x2 + x = 0 in Zp are 0 and p − 1.
2.2.4. Find all a in Z5 for which the equation ax = 1 has a solution.
2.2.5. Prove that there is no ordering ≺ of Zn such that
(i)
(ii)
if a ≺ b, and b ≺ c, then a ≺ c;
if a ≺ b, then a + c ≺ b + c for every c ∈ Zn
.
Problems from §2.3
2.3.1 If n is composite, prove that there exists a, b ∈ Zn such that a 6= [0] and b 6= [0] but ab = [0].
2.3.2 Let p be prime and assume that a 6= 0 in Zp . Prove that for any b ∈ Zp , the equation ax = b has a
solution.
2.3.3. Let a 6= [0] in Zn . Prove that ax = [0] has a nonzero solution in Zn if and only if ax = [1] has no
solution.
2.3.4. Solve the following equations.
(a) 12x = 2 in Z19 .
(b) 7x = 2 in Z24 .
(c) 31x = 1 in Z50 .
(d) 34x = 1 in Z97 .
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Homework Set 4
(Problems from Chapter 3)
§3.1
3.1.1. The following subsets of Z (with ordinary addition and multiplication) satisfy all but one of the
axioms for a ring. In each case, which axiom fails.
(a) The set S of odd integers.
(b) The set of nonnegative integers.
3.1.2.
(a) Show that the set R of all multiples of 3 is a subring of Z.
(b) Let k be a fixed integer. Show that the set of all multiples of k is a subring of Z.
3.1.3. Let R = {0, e, b, c} with addition and multiplication defined by the tables below:
+ 0
0 0
e e
b b
c c
e
e
0
c
b
b
b
c
0
e
·
0
e
b
c
c
c
b
e
0
0
0
0
0
0
e
0
e
b
c
b
0
b
e
c
c
0
c
c
0
Assume distributivity and associativity and show that R is a ring with identity. Is R commutative?
3.1.4. Let F = {0, e, a, b} with addition and multiplication defined by the tables below:
+
0
e
a
b
0
0
e
a
b
e
e
0
b
a
·
0
e
a
b
a b
a b
b a
0 e
e 0
0
0
0
0
0
e
0
e
a
b
a b
0 0
a b
b e
e a
Assume distributivity and associativity and show that R is a field.
3.1.5. Which of the following five sets are subrings of M (R). Which ones have an identity?
(a)
A=
(b)
B=
a
0
0
0
(c)
C=
(d)
D=
(e)
D=
|r∈Q
b
c
a
b
r
0
| a, b, c ∈ Z
a
b
| a, b ∈ R
a
a
0
0
a
0
0
a
|a∈R
|a∈R
3.1.6. Let R and S be rings. Show that the subset R̄ = {(r, 0S ) | r ∈ R} is a subring of R × S.
3.1.7 If R is a ring, show that R∗ = {(r, r) | r ∈ R} is a subring of R × R.
7
3.1.8. Is {1, −1, i, −i} a subring of C?
3.1.9. Let p be a positive prime and let R be the set of all rational numbers that can be written in the form
r
pi with r, i ∈ Z. Show that R is a subring of Q.
3.1.10. Let T be the ring of continuous functions from R to R and let f, g be given by
0
if x ≤ 2
2−x
if x ≤ 2
f (x) =
,
g(x) =
x−2
if 2 < x
0
if 2 < x
.
Show that f, g ∈ T and that f g = 0T , and therefore that T is not an integral domain.
3.1.11. Let
n
o
√
√
Q( 2) = r + s 2 | r, s ∈ Q
.
√
Show that Q( 2) is a subfield of R.
3.1.12. Prove Theorem 3.1: If R and S are rings, then the Cartesian product R × S can be given the
structure of a ring by setting
(r, s) + (r0 , s0 )
=
(r + r0 , s + s0 )
(r, s)(r0 , s0 )
=
(rr0 , ss0 )
0R×S
=
(0R , 0S ) .
Also, if R and S are both commutative, then so is R × S; and if R and S each have an identity, then so
does R × S.
3.1.13. Prove or disprove: If R and S are integral domains, then R × S is an integral domain.
3.1.14. Prove or disprove: If R and S are fields, then R × S is a field.
§3.2
3.2.1. If R is a ring and a, b ∈ R then
(a) (a + b)(a − b) =?
(b) (a + b)3 =?
(c) What are the answers to (a) and (b) if R is commutative?
3.2.2. An element e of a ring R is said to be idempotent if e2 = e.
(a) Find four idempotent elements of the ring M2 (R).
(b) Find all idempotents in Z12 .
3.2.3. Prove that the only idempotents in an integral domain R are 0R and 1R .
3.2.4. Prove or disprove: The set of units in a ring R with an identity is a subring of R.
3.2.5. (a) If a and b are units in a ring R with identity, prove tha ab is a unit and (ab)−1 = b−1 a−1 .
(b)Give an example to show that if a and b are units, then (ab)−1 may not be the same as a−1 b−1 . (Hint:
consider the matrices i and k in the quaterion ring H.)
3.2.6. Prove that a unit in a commutative ring cannot be a zero divisor.
3.2.7.
(a) If ab is a zero divisor in a commutative ring R, prove that a or b is a zero divisor.
(b) If a or b is a zero divisor in a commutative ring R and ab 6= 0R , prove that ab is a zero divisor.
8
3.2.8. Let S be a non-empty subset of a ring R. Prove that S is a subring if and only if for all a, b ∈ S,
both a − b and ab are in S.
3.2.9. Let R be a ring with identity. If there is a smallest integer n such that n1R = 0R , then n is said to have
characteristic n. If no such n exists, R is said to have characteristic zero. Show that Z has characteristic
zero, and that Zn has characteristic n. What is the characteristic of Z4 × Z6 ?
§3.3
3.3.1. Let R be a ring and let R∗ be the subring of R × R consisting of all elements of the form (a, a),
a ∈ R. Show that the function f : R → R∗ given by f (a) = (a, a) is an isomorphism.
3.3.2. If f : Z → Z is an isomorphism, prove that f is the identity map.
3.3.3. Show that the map f : Z → Zn given by f (a) = [a] is a surjective homomorphism but not an
isomorphism.
3.3.4. If R and S are rings and f : R → S is a ring homomorphism, prove that
f (R) = {s ∈ S | s = f (a) for somea ∈ R}
is a subring of S
3.3.5.
(a) If f : R → S and g : S → T are ring homomorphisms, show that g ◦ f : R → T is a ring homomorphism.
(b) If f : R → S and g : S → T are ring isomorphisms, show that g ◦ f : R → T is also a ring isomorphism.
3.3.6. If f : R → S is an isomorphism of rings, which of the following properties are preserved by this
isomorphism? Why?
(a) a ∈ R is a zero divisor.
(b) R is an integral domain.
(c) R is a subring of Z.
(d) a ∈ R is a solution of x2 = x.
(e) R is a ring of matrices.
3.3.7. Use the properties that are preserved by ring isomorphism to show that the first ring is not isomorphic
to the second.
(a) E (the set of even integers) and Z.
(b) R × R × R × R and M2 (R).
(c) Z4 × Z14 and Z16 .
(d) Q and R.
(e) Z × Z2 and Z.
(f) Z4 × Z4 and Z16 .
9
Homework Set 5
(Problems from Chapter 4)
§4.1
4.1.1. Perform the indicated operations in Z6 [X] and simply your answer.
(a) (3x4 + 2x3 − 4x2 + x − 4) + (4x3 + x2 + 4x + 3)
(b) (x + 1)3
4.1.2. Which of the following subsets of R[x] are subrings of R[x]? Justify your answer.
(a) S = {All polynomials with constant term 0R }.
(b) S = {Alll polynomials of degree 2 }.
(c) S = {All polynomials of degree ≤ k ∈ N, where 0 < k}.
(d) S = {All polynomials in which odd powers of x have zero coefficients}.
(e) S = {All polynomials in which even powers of x have zero coefficients}.
4.1.3. List all polynomials of degree 3 in Z2 [x].
x3 , x3 + 1, x3 + x, x3 + x + 1, x3 + x2 , x3 + x2 + 1, x3 + x2 + x, x3 + x2 + x + 1
4.1.4. Let F be a field and let f (x) be a non-zero polynomial in F [x]. Show that f (x) is a unit in F [x] if
and only if deg f (x) = 0.
§4.2
4.2.1. If a, b ∈ F and a 6= b, show that x + a and x + b are relatively prime in F [x].
4.2.2. Let f (x), g(x) ∈ F [x]. If f (x) | g(x) and g(x) | f (x), show that f (x) = cg(x) for some non-zero c ∈ F .
(b) If f (x) and g(x) are monic and f (x) | g(x) and g(x) | f (x), show that f (x) = g(x).
4.2.3. Let f (x) ∈ F [x] and assume f (x) | g(x) for every nonconstant g(x) ∈ F [x]. Show that f (x) is a
constant polynomial.
4.2.4. Let f (x), g(x) ∈ F [x], not both zero, and let d(x) = GCD (f (x), g(x)). If h(x) is a common divisor
of f (x) and g(x) of highest possible degree, then prove that h(x) = cd(x) for some nonzero c ∈ F .
4.2.5. If f (x) is relatively prime to 0F , what can be said about f (x).
4.2.6. Let f (x), g(x), h(x) ∈ F [x], with f (x) and g(x) relatively prime. If f (x) | h(x) and g(x) | h(x), prove
that f (x)g(x) | h(x).
4.2.7. Let f (x), g(x), h(x) ∈ F [x], with f (x) and g(x) relatively prime. If h(x) | f (x), prove that h(x) and
g(x) are relatively prime.
4.2.8. Let f (x), g(x), h(x) ∈ F [x], with f (x) and g(x) relatively prime. Prove that the GCD of f (x)h(x)
and g(x) is the same as the GCD of h(x) and g(x).
10
§4.3
4.3.1 Prove that f (x) and g(x) are associates in F [x] if and only if f (x) | g(x) and g(x) | f (x).
4.3.2 Prove that f (x) is irreducible in F [x] if and only if its associates are irreducible.
4.3.3. If p(x) and q(x) are nonassociate irreducibles in F [x], prove that p(x) and q(x) are relatively prime.
§4.4
4.4.1. Verify that every element of Z3 is a root of f = x3 − x ∈ Z3 .
4.4.2. Use the Factor Theorem to show that f = x7 − x factors in Z7 as
f = x (x − [1]7 ) (x − [2]7 ) (x − [3]7 ) (x − [4]7 ) (x − [5]7 ) (x − [6]7 )
.
4.4.3. If a ∈ F is a nonzero root of
f = cn xn + . . . + c1 x + c0 ∈ F [x] ,
show that a−1 is a root of
g = c0 xn + c1 xn−1 + · · · + cn
.
4.4.4. Prove that x2 + 1 is reducible in Zp [x] if and only if there exists integers a and b such that p = a + b
and ab ≡ 1 (mod p).
4.4.5. Find a polynomial of degree 2 in Z6 [x] that has four roots in Z6 . Does this contradict Corollary 4.13?