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Math 4441/6441 (Spring 2008) Test #1 (02/21) Test Problems
Problem 1 (10 points). Let X = {1, 2, 3}. Consider SX , which
consists of 6 bijectivefunctions from X to X as listed below
f1 = 1X :1 1, 2 2, 3 3; f2 :1 1, 2 3, 3 2; f3 :1 2, 2 1, 3
3;
f4 :1 2, 2 3, 3 1; f5 :1 3, 2 1, 3 2; f6 :1 , 2 , 3 .
(1) Fill in the blanks for f6 above. Make sure that f1, f2, f3,
f4, f5, f6 are distinct.(2) Determine f2 f3 and f3 f2. (That is, f2
f3 = f?, f3 f2 = f?.)(3) Determine f4 f4 and f4 f4 f4. (That is, f4
f4 = f?, f4 f4 f4 = f?.)(4) Determine f13 and f
1
4 . (That is, f1
3 = f?, f1
4 = f?.)(5) True or false (no justification needed): For
arbitrary functions g, h : X X (not
necessarily bijective), ifg f5 = h f5, then g = h. . . . . . . .
. . . . . . . . . True False
Problem 2 (10 points). In each of the following cases, determine
whether it is a function.In case of a function, determine whether
it is injective but not surjective, surjective but notinjective,
bijective, or none of the above. Show your reasoning.
(1) f : R R such that f(x) = 3x + 24.(2) g : R R such that g(x)
= x2 100.(3) h : R R such that h(x) = tanx.(4) u : R R such that
u(x) = ln(x2 + 1).(5) v : R R such that v(x) = 100x.
Problem 3 (10 points). True or false with reasoning. Let X= be a
set and letP(X) ={all subsets ofX}. Consider (binary) operations
and onP(X): for A,B P(X), defineA B = A B and A B = AB. (If you
claim Ei exists, then identify it explicitly.)
(1) There exists a (special) E1 P(X) such that E1 A = E1 for all
A P(X).(2) There exists a (special) E2 P(X) such that A E2 = A for
all A P(X).(3) The operation is commutative, i.e., A B = B A for
all A,B P(X).(4) There exists a (special) E3 P(X) such that E3A =
AE3 = A for all A P(X).
(5) There exists a (special) E4 P(X) such that E4A = AE4 = E4
for all A P(X).
Problem 4 (2+3+2+3 points). Let X,Y,Z be nonempty sets and let
f, f1, f2 : X Y,g, g1, g2 : Y Z be functions.
(1) True or false: Ifg f1 = g f2, then f1 = f2. . . . . . . . .
. . . . . . . . . . . . . . True False(2) If you chose true in (1),
prove it. If false, give a counterexample.(3) True or false: Ifg1 f
= g2 f, then g1 = g2. . . . . . . . . . . . . . . . . . . . . . .
True False(4) If you chose true in (3), prove it. If false, give a
counterexample.
Problem 5 (3+3+4 points). Let X, Y be nonempty sets and let f :
X Y and g : Y Xbe functions. Assume g f = 1X .
(1) True or false with reasoning: g is necessarily 1-1.(2) True
or false with reasoning: g is necessarily onto.(3) If, furthermore,
g is bijective, then show f g = 1Y and, consequently, f = g1.
For solutions, click here.
Math 4441 students need to do any four out of the five
problems.Math 6441 students need to do all five problems.
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