Abstract Algebra (MATS2103)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2016
MATS 203: ABSTRACT ALGEBRA
TUTORIAL SERIES V
1.
Definition: Let :
be a function defined by
number of positive integers less
than n and
relatively prime to n. is know as the Euler - function e. g. (2) =1, (3)=2, (4) = 2, (5)= 4, (6)=2.
For each of the numbers (a) n = 12, (b) n = 16, (c)n = 20, show that
l( 1mod 2),
l( 1mod
16),
( 1mod 2)
2. Find the composite mappings fog and gof, where f: and g : arc defined by
(i) f (x) = 3x
2
+2 and g(x)= 4(x+ l)
2
- 3, (ii) f(x)=
, x1 and g(x)=
3. Determine whether each of the following mappings is (i) injective, (ii) surjective; . (iii) bijective.
(a) f: , f(x)= x +3, (b) f: , f(x)= 5x -7, (c) f:
(d) f: !" ∞#, f(x)= 1 +x
3
(i) Find the domain and range of the real-valued functions of a real variable defined by
(a)= f:
(b) g(x)=
x $"
(ii) Determine the domain of the real-valued function of a teal variable, defined by g:x%
&
'
(
4. The function f: is defined by f(x) =
. Show that / is neither surjective nor injective. Find the
range of f.
6. Show that (i) f: (0, ∞ , f(x)= x
2
+ 9 is not surjective but injective.
(ii)
g: !4,∞# g(x)= x
2
+ 4 is surjective but not injective.
(iii)
h:
" ∞
1,∞ h(x)=
has an inverse and find h
-1
.
7. Let f : A B be a mapping and let X and Y be subsets of .4. Prove that
(i) f is surjective ) f(A) = B, (ii)X * Y =>f(X) * f(Y) (iii) f(X)-f(Y)*f(X - Y) with equality when J is
injective.
8. Let f : A B be a mapping and let. X and Y be subset of B, Prove that
(i) f
-l
( X u Y)
=
f
-l
(x) u f f
-l
(y), (ii) f
-l
(X-Y)= f
-l
(X) - f
-l
(Y).
9. let f : A B and f : B C be mappings. Show that «
(i)
if gof is injective, then f is injective.
(ii)
if gof is surjective, then g is injective.
(iii) if gof is surjective and g is injective, then f is surjective.
10. Let f : A B be a mapping and let {B
i
}
iEI
be a family of subsets of B; {A
i
} a of subsets of ,1. Show that
(i) f
-1
(+
,-.
/
,
=+
,-.
f
-1
/
,
(ii) f
-1
(
,-.
/
,
=
,-.
f
-1
/
,
(iii) f(
,-.
0
,
=
,-.
f0
,
(iv) f(
,-.
0
,
*
,-.
f/
,
END
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