PROVING TECHNIQUES FUNCTIONS (x,y) + + +
f(x) CO
y
=
domain
*
fix - Y
Logic pro/Direct
.
1
prog w
↳
codomain/range
·
image
↳ Proof by example/counterexample
distriction case has
ev
each current none , one or ·
by coveradiction incoming
several arrows
↳ each element has
↳
Proof by
-
·
w
exactly outgoinga
.
one
-
image of
f
5
Pigeon-hole principle G Note :
:
.
~ f(A)EY
↳ TH If
F(A) (f(x) XAy
&
iteves are containers
i put image of
·
: into A= X w
me
A under 7
: = : ·
= ,
andM > m ther there must be a
,
⑧
composition of functions Vassoc . X commutative · w
container that contains more than 1 item
. (n
fixe Mixez h(x =g(x g Ex
=
= ,
G .
Pro by induction : ·
u
reduction "r"
-
f injective X+
y ) f(x) + f(y)
-
one · : =
(
Base case
surjective FyzY 5xEX With #(x) y
: ·
=...
-
-
·
:
...
=
Inductive Step Let R
bijective fij Prop IX 141
-
:
7 f sur =
· : =
...
. + :
Induction (iH)
Mypothesis Claire
for k (
: -
function
-
·
reverse : ·
claim hords
grove for + 1
:
7 XBY
bij
. = 71 YeX with f-1 (y) (X unique)
...
: =
X
.
Proof by (strong) induction : ·
s
7
inductions "n"
~
or RELATIONS RouX and Y subset R=XXY
-
=>
Base case : n >
-
every function is a relation
p
-
=... ·
Inductive
Concatenation
-
step : Let ·
REXXY
it Claire holds k' with 1k'k
for all
-
:
---
claim hords
grove for + 1
--
T =
ROS
· B
reflexive XRX
·
SETS ·
# XeX with XRX -
·
XIY : NEX = XEY
erreflexive X Y Z
symmetric XRy yRX
⑧
)
-
=
·
X=Y :
XY1Y &X Th
·
antisyunextric xRynyRX x =
y
& =
Y
=
·
xUY =
[z/zEX VzEY] transitive
XRy1yRz => XRz
·
X1y
· =
Ez/zEX 1 zEYy (x Y)ERY or
G(y)x)
1 ·
·
inverse relation R = :
i
·
XIY setrinus relation
R=
equivalence if
-
· :
·
P(X) 2x
powerset (0-4) transitive
reflexive + syne
, .
+ -
·Wi ; X
R[X] [yeX XRy3
M
class or
equivalence of
· :
=
+ X ·
i (Visassociative
R[X] = 0 -
·
Xn(yUz) (xny)U(X 1z) =
-
·
XU(ynz) (X uy) n (X Uz) = ORDERINGS
De Laws :
Morgan In
· ·
·
XI(AUB) =
(XIA)n(X(B) ·
relation one X
:
+
antisyme .
- transitive
f(x) CO
y
=
domain
*
fix - Y
Logic pro/Direct
.
1
prog w
↳
codomain/range
·
image
↳ Proof by example/counterexample
distriction case has
ev
each current none , one or ·
by coveradiction incoming
several arrows
↳ each element has
↳
Proof by
-
·
w
exactly outgoinga
.
one
-
image of
f
5
Pigeon-hole principle G Note :
:
.
~ f(A)EY
↳ TH If
F(A) (f(x) XAy
&
iteves are containers
i put image of
·
: into A= X w
me
A under 7
: = : ·
= ,
andM > m ther there must be a
,
⑧
composition of functions Vassoc . X commutative · w
container that contains more than 1 item
. (n
fixe Mixez h(x =g(x g Ex
=
= ,
G .
Pro by induction : ·
u
reduction "r"
-
f injective X+
y ) f(x) + f(y)
-
one · : =
(
Base case
surjective FyzY 5xEX With #(x) y
: ·
=...
-
-
·
:
...
=
Inductive Step Let R
bijective fij Prop IX 141
-
:
7 f sur =
· : =
...
. + :
Induction (iH)
Mypothesis Claire
for k (
: -
function
-
·
reverse : ·
claim hords
grove for + 1
:
7 XBY
bij
. = 71 YeX with f-1 (y) (X unique)
...
: =
X
.
Proof by (strong) induction : ·
s
7
inductions "n"
~
or RELATIONS RouX and Y subset R=XXY
-
=>
Base case : n >
-
every function is a relation
p
-
=... ·
Inductive
Concatenation
-
step : Let ·
REXXY
it Claire holds k' with 1k'k
for all
-
:
---
claim hords
grove for + 1
--
T =
ROS
· B
reflexive XRX
·
SETS ·
# XeX with XRX -
·
XIY : NEX = XEY
erreflexive X Y Z
symmetric XRy yRX
⑧
)
-
=
·
X=Y :
XY1Y &X Th
·
antisyunextric xRynyRX x =
y
& =
Y
=
·
xUY =
[z/zEX VzEY] transitive
XRy1yRz => XRz
·
X1y
· =
Ez/zEX 1 zEYy (x Y)ERY or
G(y)x)
1 ·
·
inverse relation R = :
i
·
XIY setrinus relation
R=
equivalence if
-
· :
·
P(X) 2x
powerset (0-4) transitive
reflexive + syne
, .
+ -
·Wi ; X
R[X] [yeX XRy3
M
class or
equivalence of
· :
=
+ X ·
i (Visassociative
R[X] = 0 -
·
Xn(yUz) (xny)U(X 1z) =
-
·
XU(ynz) (X uy) n (X Uz) = ORDERINGS
De Laws :
Morgan In
· ·
·
XI(AUB) =
(XIA)n(X(B) ·
relation one X
:
+
antisyme .
- transitive
