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(MATH341)[2003](s)midterm~wfli^_10507.pdf
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FINAL EXAM -MATH 341
Monday, April 28, 2003
INSTRUCTOR: George Voutsadakis
This is a true/false exam. Read each question very carefully before an-swering. Each question is worth 1 point.
RULES:
. Rule Half a point will be subtracted for each wrong answer.
+ Rule If you have at least 60 out of the 90 questions answered correctly, 10 more points will be added to your score for free!!
GOOD LUCK!!
1. (a) i.If f : A B and g : B A are such that g . f =1A, then g must be 1-1.
ii. If f : A B is 1-1 and g : B C is onto, then g . f is onto.
iii. f : A B is said to be invertible if there exists a g : B A such that g . f =1A.
iv. f : A B is invertible if and only if A and B have the same cardinality.
v. |Z| = |nZ|.
(b) i. A relation on a set A is a subset of A.
ii. A relation R . A A is re.exive if, for all a, b A, aRb implies bRa.
iii. If R is an equivalence relation on A, the equivalence class of a A is [a]= {b A : aRb}.
iv.
In IR2 , the relation de.ned by (x1,y1) (x2,y2) if and only if x1y2 = x2y1 is an equivalence relation.
v.
Given a function f : A B, the relation on A de.ned by a b if and only if f(a)= f(b), for all a, b A, is an equivalence relation.
2. (a) i. A set with n elements has 2n.1 subsets.
ii. Given any integers a and b 1, there exist unique integers q, r, such that a = qb + r.
iii. Two integers a and b are relatively prime if their only positive common divisor is 1.
iv.
Let a, b, c be integers. If c\ab, then c\a or c\b.
v.
Forany[r] U(n), there is an [s] U(n), such that [r][s] = [1].
(b) i. The polar representation of .i is .1(cos + i sin ).
22
ii. Given any complex number z = r(cos + i sin ), for any positive integer n, we have
n
z= rn(cos n + i sin n).
1
iii. The inverse of z = 2+3i, 1 = is not a complex number because it is not written z 2+3i 1
in the form a + bi, with a, b IR.
iv. The complex conjugate of .1 + 7i is .2 + 1 i.2 7
v. The equation z5 + 8i = 0 has 5 complex roots.
3. (a) i. A matrix A M(n, IR) is invertible i. there exists a matrix B M(n, IR), such
that A + B = B + A = 0.
ii. There exist matrices A, B M(2, IR), such that AB = I2 but BA .= I2.
iii. For any A, B M(2, IR), we have det(AB) = det(A)det(B). c d b iv. If A = is invertible, then det(A) .= 0 and A.1 = . a b . 1 . .a det(A) c .d . .
v. In M(2, Z7), there is no matrix with determinant equal to 2.
(b) i. The n-th complex roots of unity form a group of order n . 1 under complex multi-
plication.
ii. A group .G, .. is commutative if and only if a . (b . c) = (b . c) . a, for all a, b, c G.
iii. The dihedral group Dn has order 2n, if n is even and 2n . 2 if n is odd.
iv. The rational numbers Q form a group under multiplication.
v. Let be the Euler -function. Then, for all primes p and positive integers k,
(pk) = pk(1 . 1 ).p
4. (a) i. A nonempty