(viii) If X is an innite set, then the family of all nite subsets of X
forms a subring of the Boolean ring B( X ).
3.2 Prove that a commutative ring R has a unique one 1; that is, if e R
satises er = r for all r R , then e = 1.
Prove that 10q + r is divisible by 7 if and only if q 2r is divisible
Solution. If 10q + r 0 mod 7, then 15q + 5r 0 mod 7,
and so q 2r 0 mod 7. Conversely, if q 2r 0 mod 7,
then 3q 6r 0 mod 7, hence 3q + r 0 mod 7, and so
10q + r 0 mod 7
(iii) Prove that the ring of Gaussian integers is a domain.
Solution. The set Gaussian integers form a subring of C, and
hence it is a domain.
3.12 Prove that the intersection of any family of subrings of a commutative ring
R is a subring of R .
A similar argument shows that if pe | pr i for i 1, then pe | pr m i .
By the fundamental theorem of arithmetic, the total number of factors p
occurring on each side must be the same. Therefore, the total number of
p s dividing apr must equal the total
2.12 Let X = cfw_x1 , . . . , xm and Y = cfw_ y1 , . . . , yn be nite sets, where the xi are
distinct and the y j are distinct. Show that there is a bijection f : X Y if
and only if | X | = |Y |; that is, m = n .
Solution. The hint is essentially the
Write x 15 1 as a product of irreducible polynomials in F2 [x ].
(ii) Find an irreducible quartic polynomial g (x ) F2 [x ], and use it to
dene a primitive 15th root of unity F16 .
(iii) Find a BCH code C o
(ii) If f : R R is a bijection whose graph consists of certain points
(a , b) [of course, b = f (a )], prove that the graph of f 1 is
cfw_(b, a ) : (a , b) f .
Solution. By denition, f 1 (b) = a if and only if b = f (a ).
Hence, the graph of f 1 consis
(ii) Prove that
| Br (u )| =
(q 1)i .
If C An is an (n , M , d )-code,
(iii) (Gilbert-Varshamov bound.)
where |A| = q , prove that
d 1 n
i =0 i
4.70 (Hamming bound)If C An
2.5 Let A and B be sets, and let a A and b B . Dene their ordered pair
(a , b) = cfw_a , cfw_a , b.
If a A and b B , prove that (a , b ) = (a , b) if and only if a = a and
b = b.
Solution. The result is obviously true if a = a and b = b.
Solution. If Av = cv , then Am v = cm v for all m 1. Hence, if Am = 0,
then cm = 0 (because eigenvectors are nonzero). Thus, c = 0.
Conversely, if all the eigenvalues of an n n matrix A are 0, then the
characteristic polynomial is h A (x ) = (x 0) (x
There is a circular castle whose diameter is unknown; it is provided with four gates, and two lengths out of the north gate there
is a large tree, which is visible from a point six lengths east of
the south gate. What is the length of the diameter
Solution. Let z Z . Since g is surjective, there is y Y with
g ( y ) = z ; since f is surjective, there is x X with f (x ) = y . It
follows that (g f )(x ) = g ( f (x ) = g ( y ) = z , and so g f is
(iii) If both f and g are bijective, prov
(ii) Show that one root of f ( X ) = X 3 + X 2 36 is an integer and
nd the other two roots. Compare your method with Cardanos
formula and with Vi` tes trigonometric solution.
Solution. Corollary 3.91 says that any integer root is a divisor of
(ii) If B Y , prove that f 1 ( B ) = f 1 ( B ) , where B denotes the
complement of B .
2.18 Let f : X Y be a function. Dene a relation on X by x x if f (x ) =
f (x ). Prove that is an equivalence relation. If x X and f (x ) = y , the
Solution. We know that cos is a root of f (x ) = 4x 3 3x r , where
r = cos 3 . In particular, if = + 120 , then cos is a root of 4x 3
3x cos 3( + 120 ). But the addition formula for cosine gives
cos 3( + 120 ) = cos 3 = r,
and so cos( + 120 ) is also
(vii) Every transposition is an even permutation.
(viii) If a permutation is a product of 3 transpositions, then it cannot
be a product of 4 transpositions.
(ix) If a permutation is a product of 3 transpositions, then i
5.13 Find the roots of x 3 6x + 4.
Solution. We have q = 6, r = 4, D = 16, and 3 = 2 + 2i . Use De
Moivres theorem to nd cube roots of 2 + 2i :
(1 + i )3 = 2 + 2i .
Hence, = 1 + i , = 1 i , and a root of the polynomial is + = 2.
The other two roots ca
(ii) If kr n , where 1 < r n , prove that the number of permutations Sn , where is a product of k disjoint r -cycles, is
k ! r k [n (n
1) (n kr + 1).]
If is an r -cycle, show that r = (1).
Solution. If = (i 0 . . . ir 1 )
Thus, m = 2, = 3, and we have the factorization
x 4 15x 2 20x 6 = (x 2 + 4x + 3)(x 2 4x 2).
The quadratic formula applied to each of the two factors gives the desired
3, 1, 2 + 6, 2 6.
In particular, all the roots are real.
Remark. Given the di
2.2 If A and B are subsets of a set X , prove that A B = A B , where
B = X B is the complement of B .
Solution. This is one of the beginning set theory exercises that is so easy
it is difcult; the difculty is that the whole proof turns on the meaning o
(iv) Prove that every hermitian matrix A over C is diagonalizable.
4.55 If A is an m n matrix over a eld k , prove that rank( A) d if and
only if A has a nonsingular d d submatrix. Conclude that rank( A) is the
maximum such d .
1.102 My Uncle Ben was born in Pogrebishte, a village near Kiev, and he claimed
that his birthday was February 29, 1900. I told him that this could not be,
for 1900 was not a leap year. Why was I wrong?
Solution. Even though 1900 was not a leap year in
(iii) Gaussian equivalent matrices have the same column space.
(iv) The matrix A =
0 1 1
(v) Every nonsingular matrix over a eld is a product of elementary
(vi) If A
(iv) For all rationals a , b, c, prove p (a , b) p (a , c) + p (c, b).
Solution. p (a , b) p (a , c) + p (c, b) because
p (a , b) = a b p = (a c) + (c b)
maxcfw_ a c p , c b p
ac p + cb p
= p (a , c) + p (c, b).
(v) If a and b are integers and pn
(ii) Prove that lies in the column space of A if and only if rank([ A| ])
= rank( A).
Solution. Let A be Gaussian equivalent to an echelon matrix U ,
so that there is a nonsingular matrix P with P A = U . Then lies
in the row space Row( A) if and only
c = b = db , so that a m = b . Thus, a divides b ; as
(a , b ) = 1, we have a divides , by Corollary 1.40. Write =
a k , and observe that
c = db
= db a k = (db )(da )k /d = [ab/d ]k .
Therefore, ab/d = [a , b], and so [a , b](a , b) = ab.
(ii) Find [13
(ii) Every matrix over R is similar to innitely many different matrices.
(iii) If S and T are linear transformations on the plane R2 that agree on
two nonzero points, then S = T .
(iv) If A and B are n n nonsingular ma
(ii) Prove that integration S : V3 V4 , dened by S ( f ) = 0 f (t ) dt ,
is a linear transformation, and nd the matrix A = X 4 [ S ] X of in3
4.34 If Sn and P = P is the corresponding permutation matrix, prove that
P 1 =
prove that there exists a unique solution if and only if det( A) = 0.
(ii) If V is a vector space with basis X = v1 , v2 , dene T : V V
by T (v1 ) = a v1 + bv2 and T (v2 ) = cv1 + d v2 . Prove that T is a
nonsingular linear transform