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O'Reilly Platform
book
The Nuts and Bolts of Proofs, 4th Edition
by
Antonella Cupillari
January 2011
Beginner content level
Beginner
296 pages
11h 43m
English
Academic Press
Content preview from
The Nuts and Bolts of Proofs, 4th Edition
Let
x
∈
⋃
f
∈
F
A
f
′
.
B y
definiti on
of
comple
ment,
this
implies
that
x
∉
⋃
f
∈
F
A
f
.
By
definition
of
union,
the
fact
that
an
element
does
not
belong
to
the
union
implies
that
the
element
fails
to
belong
to
e very
one
of
the
sets
us ed
in
t he
union.
So,
x
∉
⋃
f
∈
F
A
f
implies
that
x
∉
A
f
for
all
f
∈
F
.
By
definition
of
complement
of
a
set,
this
is
equivalent
to
stating
that
x
∈
A
′
f
for
all
f
∈
F
.So
x
∈⋂
f
∈
F
A
′
f
.
Therefore
⋃
f
∈
F
A
f
′
⊆⋂
f
∈
F
A
′
f
.
Second
part:
⋃
f
∈
F
A
f
′
⊇⋂
f
∈
F
A
′
f
.
Let
x
∈⋂
f
∈
F
A
′
f
.
By
definition
of
intersection,
the
element
x
will
be
in
the
intersection
o nly
if
it
belongs
to
every
set
in
the
int ersection.
By
definitio n
of
complement,
this
imp ...
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Publisher Resources
ISBN: 9780123822178