Exercise Sheet 1
1)
Let A, B be sets. What does the statement "A is not a subset of B " mean?
2)
Let A, B, C, X be sets with A, B, and C are subsets of X. Prove the following set equalities
a) (A
∩
B)' = A'
∪
B'
b) (A
∪
B)' = A'
∩
B'
where the complements are taken in X.
c) A \ B = A
∩
B'
d) A
∩
(B \ C) = (A
∩
B)\(A
∩
C)
3)
Let A = {x
∈
R
 x
2
5x+4
≤
0}; B = {x
∈
R

x
2
 1/2
≤
1} and C = {x
∈
R
 x
2
7x+12 <
0}.
Determine (A
∪
B)
∩
C.
4)
Let
f: X
→
Y be a mapping; Let A, B
⊂
X; C, D
⊂
Y. Prove that :
a) f(A
∩
B)
⊂
f(A)
∩
f(B); Find examples of A, B, and f such that
f(A
∩
B)
≠
f(A)
∩
f(B)
b) f
1
(C
∩
D)
=
f
1
(C)
∩
f
1
(D)
5)
Let
f :
R
\ {0}
→
R
and
g :
R
→
R
x

→
1/x
x 
→
2x/(1+x
2
)
be mappings.
a) Determine
f
◦
g and
g
◦
f.
b) Find the image g(
R).
Is g injective? surjective? (Answer the same question for f. )
6)
Let f :
A
→
C and g : B
→
D be two mappings. Consider the mapping h : A
x
B
→
C
x
D
defined by h(a, b) = (f(a), g(b)) for all (a, b)
∈
A
x
B.
a) Prove that, f and g are both injective if and only if h is injective.
b) Prove that, f and g are both surjective if and only if h is surjective.
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Exercise Sheet 2
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 Spring '09
 haiza
 Linear Algebra, Vector Space, Tn, 2m, Cramer, EXERCISE SHEET

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