6.042/18.062J
Mathematics
for
Computer
Science
September
10,
2010
Tom
Leighton
and
Marten
van
Dijk
Notes
for
Recitation
1
1
Logic
How
can
one
discuss
mathematics
with
logical
precision,
when
the
English
language
is
itself
riddled
with
ambiguities?
For
example,
imagine
that
you
ask
a
friend
what
kind
of
dessert
was
offered
at
the
party
you
couldn’t
make
it
to
last
week,
and
your
friend
says,
You
could
have
cake
or
ice
cream.
Does
this
mean
that
you
could
have
both
cake
and
ice
cream?
Or
does
it
mean
you
had
to
choose
either
one
or
the
other?
To
cope
with
such
ambiguities,
mathematicians
have
defined
precise
meanings
for
some
key
words
and
phrases.
Furthermore,
they
have
devised
symbols
to
represent
those
words.
For
example,
if
P
is
a
proposition,
then
“not
P
”
is
a
new
proposition
that
is
true
whenever
P
is
false
and
vice
versa.
The
symbolic
representation
for
“not
P
”
is
P
or
P
.
¬
Two
propositions,
P
and
Q
,
can
be
joined
by
“and”,
“or”,
“implies”,
or
“if
and
only
if”
to
form
a
new
proposition.
The
truth
of
this
new
proposition
is
determined
by
the
truth
of
P
and
Q
according
to
the
table
below.
Symbolic
equivalents
are
given
in
parentheses.
“
P
implies
Q
”
or
“
P
if
and
only
if
Q
”
or
“
P
and
Q
”
“
P
or
Q
”
“if
P
,
then
Q
”
“
P
iff
Q
”
P
F
Q
F
(
P
∧
Q
)
F
(
P
∨
Q
)
F
(
P
⇒
Q
)
T
(
P
⇔
Q
)
T
F
T
F
T
T
F
T
F
F
T
F
F
T
T
T
T
T
T
There
are
a
couple
notable
features
hidden
in
this
table:
•
The
phrase
“
P
or
Q
”
is
true
if
P
is
true,
Q
is
true,
or
both.
Thus,
you
can
have
your
cake
and
ice
cream
too.
•
The
phrase
“
P
implies
Q
”
(equivalently,
“if
P
,
then
Q
”)
is
true
when
P
is
false
or
Q
is
true
.
Thus,
“if
the
moon
is
made
of
green
cheese,
then
there
will
be
no
final
in
6.042”
is
a
true
statement.
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Recitation
1
2
There
are
two
more
important
phrases
in
mathematical
writing:
“for
all”
(symbolized
by
∀
)
and
“there
exists”
(symbolized
by
∃
).
These
are
called
quantifiers
.
A
quantifier
is
always
followed
by
a
variable
(and
perhaps
an
indication
of
the
range
of
that
variable)
and
then
a
predicate,
which
typically
involves
that
variable.
Here
are
two
examples:
x
∀
x
∈
R
+
e
<
(1
+
x
)
1+
x
∃
n
∈
N
2
n
>
(100
n
)
100
x
The
first
statement
says
that
e
is
less
than
(1
+
x
)
1+
x
for
every
positive
real
number
x
.
The
second
statement
says
that
there
exists
a
natural
number
n
such
that
2
n
>
(100
n
)
100
.
The
special
symbols
such
as
∀
,
∃
,
¬
,
and
∨
are
useful
to
logicians
trying
to
express
mathematical
ideas
without
resorting
to
English
at
all.
And
other
mathematicians
often
use
these
symbols
as
a
shorthand.
We
recommend
using
them
sparingly,
however,
because
decrypting
statements
written
in
this
symbolic
language
can
be
challenging!
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Logic, Quantification, Universal quantification, Existential quantification, Cabal

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