NUMBERS AND SETS EXAMPLES SHEET 1.
W. T. G.
1. Let
A
,
B
and
C
be three sets. Give a proof that
A
\
(
B
∪
C
) = (
A
\
B
)
∩
(
A
\
C
) using
the criterion for equality of sets.
2. The
symmetric difference
A
4
B
of
A
and
B
is defined to be (
A
\
B
)
∪
(
B
\
A
). (That
is, it is the set of elements that belong to one of
A
and
B
but not both.) Write out a truth
table to show that the operation
4
is associative. Show that
x
belongs to
A
4
(
B
4
C
) if
and only if
x
belongs to an odd number of the sets
A
,
B
and
C
and use this observation
to give a second proof that
4
is associative.
3. Let
A
,
B
,
C
and
D
be sets. Prove that
A
×
(
B
∪
C
) = (
A
×
B
)
∪
(
A
×
C
). Is it necessarily
true that (
A
×
B
)
∪
(
C
×
D
) = (
A
∪
C
)
×
(
B
∪
D
)?
4. Write down the negations of the following statements.
(i)
n
is even or
m
is a multiple of 3.
(ii) Every
x
∈
A
is also an element of
B
∩
C
.
(iii) If it is not raining today then no pigs can fly.
5. Let
f
and
g
be functions and let
h
=
g
◦
f
. If
f
and
g
are injections/surjections, prove
that
h
is an injection/surjection.
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 Fall '10
 STAFF
 Addition, Sets

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