CPSC
at
141
Lecture
Mtn
A
valid argument
the
conclusion
is
which
in
one
Notes
all
premises
true
are
well
as
as
.
general form of
The
( P ^R^R^
.nPn)
.
Pn
Definition 2.4
It
:
tautology
a
write
Basically this
is
,
both
The
To
:
Roger
0
1
is
statements
,
the
C PK
A
declarative
It
1.)
2.) It
statement
one
is
statement
not
a
3.) It
becomes
a
replaced
by
are
that
be
can
"
Example : The
The

the
universe
This
"

and
To
use
)
,
:
y
number
xt2
The
) : The
the
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or
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y
( choices
choices
even
be
,
y
integer
Cpx
In
the
last
rcpititions
Where
r
"
lecture
We
:
2 8th
September
141
"
,
Lecture Notes
talked
we
about
arranging
are
greater than
is
I
"
with
combinations
from
obiects
"
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objects
n
"
n
.
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r&f
( divide
these
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n
among
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objects
dividers )
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sol
CPSC
Ex
:
There
three
are
each
game
day
least
at
A : The
:
The
25
Anc
=
each
Using
a
there
Subtract
are
the
the
during
arrangements
by the
possibilities
each
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=
day
vein
a
game
without
week
week without
day
15
=
as
playing
over
there
diagram
days
is
.
CPSC
Lecture
141
4.6 that
combination of a and b.
From Theorem
and
a
b
exists
x
when
smallest
is the
,
prime
ax + by
where
Z
e
y
,
gcdla b)
the
relatively
are
Nor
Notes
gcdla
,
b)
I
=
,
linear
that
is
,
there
I
=
18
Example
:
( 42,70 ) 14 so 42+701=14 whi
Cpsc
141
If
Statements
two
equivalent
logically
two
Oct 5th
Example
We
the
use
of
P

p
they
,
said
are
'
to
columns
are
sign
be
to
represent
q
p
Pvq
.
elk
Notes
.
Pvq>

:
p
equivalent
are
.
statements
equivalent
Lecture
q
tin
.
We
that
should
also
notic
CPSC
Lecture
141
Notes
C.atiseankoduct.tl/B=Ha.b)/aeA.beB
Example
:
A dice
A
coin
6
side
Definition
2
,
cfw_ F
,
B
,
,
,
.
.
3,4
(6
.
5,6
,
B)
,
12
5.2
For sets A
:
relation from A
REAXB
.
is
,
B
,
any
subset
B Any
to
A
Any
C
=
Ill B) t.FI ( 2,13 )
,
ID
Cpx

In
last
our
and
lecture
went
we
,
theorem
nomial
multi
5*25*2015
Lecture Notes
141
binomial
the
over
theorem
.
tllcombinationswithoutrepititionsy

When
allowed
were
in
row
a
We
.
Example :
have four
or
is
to
each
having
to
also
have
One
"
barge
thr
CPSC
Definition 5.6
If
:
HA )=cfw_
,
HA )
and
is
,
Theorem
5.2
AIHAVAI
c.)
be
called
,
b) be
of
,
,
oo
)
5.9
:
A
)
A .=foo
,
A
,
,
A EA
.
0
will
,
,
Then
.
be HA
^
,
on
A
f
HA NAD
be
e
is injective
f( A )
]
a
NAIEHAINHAI
,
what
be
.
Something like the ab
CPSC
141 Lecture
A B C U
For
,
define
we
,
the
a.) AUB ( the union of A and B)
Bt the
b.) An
"
:
following
=cfw_ XKEAVXEB
B) =Ex/xeAnxeB
intersection of A and
As Blthe
c.)
Qt 30
Notes
symmetric difference of
B)
A and
=cfw_XkeAvxeB)nxAnB=ExlxeAUBrxe AND
Lecture Notes Na kth
CPSC 141
Theorem 4.5
unique
q
,
If
:
Z
13>0
with
,
with
2
e
r
.be
a
a
qbtr
=
,
then there
OI
,
exist
Ib
r
20=(3/4)+2
that
It
is
I
iii
to
important
note
cannot
r
be negative
3=(16)+3
Definition
4.2
be
to
said
0
Then
.
of
divisor
a.)
Statements ( or
that are
either
We
following
It
3)
that
true
The
or
.
is
2.) Get
up
for
course
"
wrote
Gone
with
the
"
'
of
but
beatital
a
:
required
a
Mitchell
the
not
following sentences
It What
to
2+2=5
each
false
letters
.
Margaret
Wild
sentences
deda
CPSC
141 Lecture Notes
4.2
Theorem
The
:
Alternative

5h )
Let
one
Form
denote
more
or
,
for
Sh )
then
1.)
2.)
n
E=O
F=
+
.
,
let
.
.
,
,
,
then
the
for
all
SG
+1 )
1)

,
,
.
,
.
but
statement
n2
.
,
no
n
,
,
(K
Z
E
with
Sh )
and
s
which
n
are
,

l
)
a
at
The
that
and
The
7+1
CPSC
,
Principal of
contain
v
.
dual
no
Lecture
141
Notes
Duality Let s and t
logical connectives other
:
It
s >
of
a
t
,
statements
be
than
^
sd , H
then
statement
simply the some
^
each
with
symbol being replaced by v and V
Note
CPSC
Lecture Notes
141
Definition 4.4 :
a
of
,
b,
both
c
a
multiple of
integers that
Theorem
4.10
:
For
all
Zt
e
,b
a
b
are
Far
if
it
a
common
a.be
20h
is called
multiple
Furthermore
.
,
c
,
Nov
I
,
c
common
a
is
a
least
common
smallest of all positive
of
CPSC 141
Lecture
Notes
6th
Nor
( 2iH=
Ex :
)
Induction
i
its
.
k41
n
1+3+5
=3
1+3+5+7
n=4
this
Claim
is
an
open
Mathematical
Using
Base
Step :
n=l
both
sides
Since
Induction
SLKH )
We
so
are
Assumption
true
is
what
to
A
get
i
:
K 't
.
2kt
I
2411=1
equal
CPSCMI
Explicit
term
Definition
no
Definition
tom
.
is
Ex :
L
,
Prove
L
By
for
the
,
n
as
Ln =L
Ln
,
=
En
know
what
We
can't
tell
the
nth
What the
!
n
=
.
(n
.
1) !
.
Consider
Ln=Fm
and
La
+
which
.
+Fk+
,
.
,
Vnezt
for
n
EZ
with
NI
are
true
both
Eti
Fed
that
Stuff
Sections
It
Vein
Where
and
0
=
Lecture
141
CPSC
be
will
1.1
diagrams
in
the
on
1.4

2.1
,
set
and
,
asked to
14
4th
exam
3.3

Nov
Lil
and
,
5.5

theory
IAI 1131
IANBKIANCHI Bnd
know
we
Notes
,
B
.
find
C
2.)
Given
of
letters
the
3.) 21 EX
ea
CPSC
The
A
of ways
number
ta
we
can
of functions
number
A=
Lecture
141
AXA
b.c d
,
,
a
b
a
b
a
b
C
D
*
a
\
b
X
X

c
d
A
a
d
c
If
10
d
If
a
b
C
d
don't
b
c
c
d
d
is
a
an
4
416
entries
these
choices
identity
,
49
functions
reflected
are
entries
doubled
ar
CPSC
Lecture
141
Notes
C.atiseankoduct.tl/B=Ha.b)/aeA.beB
Example
:
A dice
A
coin
6
side
Definition
2
,
cfw_ F
,
B
,
,
,
.
.
3,4
(6
.
5,6
,
B)
,
12
5.2
For sets A
:
relation from A
REAXB
.
is
,
B
,
any
subset
B Any
to
A
Any
C
=
Ill B) t.FI ( 2,13 )
,
ID
CPSC
Lecture Notes
141
Definition 4.4 :
a
of
,
b,
both
c
a
multiple of
integers that
Theorem
4.10
:
For
all
Zt
e
,b
a
b
are
Far
if
it
a
common
a.be
20h
is called
multiple
Furthermore
.
,
c
,
Nov
I
,
c
common
a
is
a
least
common
smallest of all positive
of
CPSC
141
Ex : Plx )
is
:
x
odd
is
statement
'
x
:
,

l
then
is
x
even
'

is
I
even
.
:
)]
9lx
Ux ) ]
[ Px

x
of this
tx[PH

2Gt
911
Negation
If
:
HE Plxl
94
odd
"
Statement
At
Lecture Notes
[ Plxlv

911 ]
[ Plxh HXD

Qualifiers
have
can
y[
kitty [
CPSC
Example
AAI
Lecture Notes
141
=
CAUBITANBI
HAUBTUCAMB
HINBTUA
nd)
De Morgan 's Law
Law of Double Complement
Demerger 's
HTANIVIHKANBWB
=[
IAUATNBUIDMTAVBTNBUBT
IIUBINAUBT
=tAUBWAnB
=
=A:
( AUBTNIUB
(
Law
)]
tun CBVAHNKAUBTNU ]

2nd
:
TVBTUIANBI
=
No
CBC
Lecture
141
Definition
5.3
For
:
f
A
exactly
IAI =3
once
in
,
,
25
Nov
B
which
in
function
a
,
or
,
element
every
first component
the
as
the
mapping
of
of
A
an
,
appears
ordered
relation
1131=4
# of relations 2*131=2
Ata
B
to
A
sets
empty
non
from
pai
CPSC
Definition 5.6
If
:
HA )=cfw_
,
HA )
and
is
,
Theorem
5.2
AIHAVAI
c.)
be
called
,
b) be
of
,
,
oo
)
5.9
:
A
)
A .=foo
,
A
,
,
A EA
.
0
will
,
,
Then
.
be HA
^
,
on
A
f
HA NAD
be
e
is injective
f( A )
]
a
NAIEHAINHAI
,
what
be
.
Something like the ab
141 Lecture
CPSC
Notes
2k
Qt
AGB
ktx
>
"
This
all
Er
says
A
If
A
it
,
E
x
A
,
then
EB
13
=
A=
If
B)
xe
A
th (
Be
x
A
H
x
13
e
)
C 13
ACB
AEB
>
Vx(xA
At B
^
13 ) n
X

th
#
(
x>
EB
x
)
#
n3xtXeA
EBHX x
ktxe
(X
e
A
x
B
(xA
ktxeA
ktxeA
e
x
Vxlxe A
x
Al
X
B)
9* CPSC
at
2.16
Example
c (
Hohn
Simplifying
statements
Notes
.
t.PH

Pvqlnt

>(Pvqh(
>Pv( qr
) Pvfo
)
:
Lecture
141
Pv
Pv

4

of )

g)
Demorgan 's
Law
Double
Law of
Negation
Distributive Law adv over ^
Inverse Low
p
Identity
Law
of
Ytdsidpeamgiuse
Oct
141
Cpsc
Natural
can
language
statement
a
Lecture Notes
2nd
confusion
cause
when
interpreting
.
Ex
P
:
:
Statement
Use
could
be
1
The
Pu
You
( Pr
have
is
true
A
,
^
I
:
r
is
all
Ex
A
and
or
compound
be
can
with
goes
confusing
what
.
9^r
confusing
orde