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Proposition a^ statement^ that^ is^ true^ or^ false
at the (^) current time X (^20) X I'm
wearing
a black (^) shirt
negation
I'm
wearing
a (^) black shirt pp (^9) proposition I'm (^) not
wearing
a (^) black shirt
Pag conjunction^ I'm^ wearing a^ black^ shirt^ and^ there
p and q (^) are 20 Students in^ class V
q disjunction^
I'm
wearing
a black shirt^ or there
p or^ q are^
20 students in class
P q
exclusive (^) or I'm
wearing
a black shirt^ or there
p exclusive^ or
q
are 20 students in class but not
both
p a^ implication If I'm^ wearing a^ black^ shirt
then there^ are^20 students^ in^ class
pas g^
biconditional^ I'm
wearing
a black^ shirt^ if
p q and^ a sp and^
only
if there^
are (^20) student in (^) class
p (^) q F pig (^) pug peg^ peg^ q op^ peg
T T
F T T T^ T T T
T F F^ F^ T
T F T F
F T^ T^
T
T T^ T^
F F
F F
T F^ F^ F^ T T^ T
Wo p (^) q F^ q era E (^) Frg
T F
F T F^ F F (^) T (^) T F (^) T T
T T^ F^ F
F F^ T^ T^ I^ T
T
p is^ a^ sufficient condition^ for g
p (^) q if^
p then a
p
is a necessary condition (^) for
q
a p if q then^ p
I F E^ g p ab^ b^ a
contrapositive P (^) a F q F^ q (^) q p T (^) T F F T T
X EZ^ x^15
F X^ EZ^ PIX^ True
F XE^ Z^ X^715 An (^) integer n is^ even if we can^ write it (^) as n 2k^ where K E (^) Z An (^) integer n is^ even if I^ KEZ sit n 2K An
integer
n is^ odd^ if we^ can^ write^ it^ as n (^) 2kt where^ K (^2) An (^) integer n is^ odd if IKE (^) Z (^) S.t n 2kt 24ft An integer
n is^ prime
if n i^ and^ for all positive integers
arb when^ a^ ban either
a I^ or^ b^ l
otherwise a^
is composite
A number n^ is rational
if it can^ be (^) written as I^ where a (^) b EZ^ and^ b^ o
otherwise n is^ irrational
rounding
down
n (^) nez and nexcutt
1 Me
rounding
up in^ if
nez and n x ntl
ex (^) if you
add 2 even numbers
you get^ an even number let x (^) be an (^) even (^) integer let
y
be an (^) even (^) integer X 2K^ where^ KE^ Z Y 2jCwherej^
for
ppl to recognize xty 2k42g Kt (^) KtjE Xty 2m where^ ME^2
ex the^ product
of 2 rational (^) numbers is
rational
let x be a rational number
let
y
be a rational number
x where^ a^ b EZ y
F
where K j EZ x y E
I
Gtfo
prove bj
EZ Obj akE^ Z^ X
Y
T where^ M NEZ
m
q X (^) TM LKTES IK^ OSE I Prove (^) that
if x^ and^
y are integers where xty is even (^) then x (^) and y are both (^) even or^ both
odd
if p then
g
p q I^75
proof (^) by contrapositive if x and y are both (^) not even^ and both not
odd then^
xty is (^) not even if one of x and y is (^) even and the (^) other is odd then key is odd
case I^ x is^ even^ and^
y
is odd
D 2K KE^ Z 4
1
JEZ
Xt 2k^
21kt ti^ Kt EZ
Kt M M^ E^ Z Xt y 2Mt (^) I
case 2 x is^ odd^ and^
y is even X 2kt^ I^ KE^ Z 2
j
ez
y Tty 2kt 2
21kt 1 kt^ E^ Z
Kt M^ MEZ Xty 2mtl
without loss of generality let^ x be^ even^ and
y be
odd
ex show^ that atleast^3 of
any 25 days chosen (^) must fall
within the same month
proof (^) by
contradiction
assume
p
is (^) false I
assume that I^ can
pick 25 days of the^ year such (^) that each month has a^ max^ of (^2) days j e 2 j ft^ d^
t I^2
all (^) days that^ are^ chosen^24
this is^ a^ contradiction^ gin^ we
chose (^25) days if p then^ q
direct start^ with^ p prove
q
contrapositive start^ with^
a proved contradiction (^) start with (^) P and (^) E prove
contradiction
I Zp^11 PEZ
min is odd
Tna EVE a IF^ AT a is even^ a is^ even a 2M MEZ 232 92 2b 2m
262 4M
D2 2m2 ME E^ Z
b is even^ b is even a is^ even^2 a b (^) is even a b a and^ b^ have^ a^ common^ factor of (^2 ) contradiction
Prove that there^ exists irrational^ number^ x
and y
such that^ x is^ rational
case 1 I is rational
X T2^ x is^ rational
4 52
case 2 T2^ is^ irrational
X
E irrational
y
F
T
T2 Z
XY is (^) rational
