Homework Set #3
1 a) (A + B)n)
((A(B)u(B(A))nC
distrative((((B)] /C (BLA)]
.
=
=> AnB)u(CBnAY
(AaB)((A)(B]u/(B)(A)
(An(a(bnc)
=
(AnCa(bnc)
same
set
(AnD(B)u/(Bnc)(A)
=
Since (AAB)1C
the
(And)(Brc) u(brc))(Anc)
=
,
(((A)B]r(((13) (A)
and
they
(An()s (BMC)
are
equal
.
are
both
equal to
1 b) (Aab)(
(AAB)nC
=
.
=
((A(B)u(B(A)] 1C
tributive
(AAB)
=
(Al)y (BIC)
((((B)(U/Ch (B(A)
(nAnBu/inBra
/(A)() ( (Bic)( v/(sic) ((A())
=
= Ancin(snctYulBnc'n(Ance)
(Anc(Buc))u(Brcn(Auc))
#
(An(nB(inc)u(B(A)u(n))
Entradict
(An(nB9Uf) u(Bn((a)) uf)
=> AncinByu/BnCAY
(Ally (b(c)
=
Since
(AsB)/C
same
Set
,
they
(nAnBYu/cnBnA
and
are
(AIC)1 (BIC)
.
equal
are
both equal
to the
2
.
M
a)
P
C
=
"Mary
"
=
=
she
"She
textbook
sells her
can
get
a
the
passes
"
good price"
class
"
M = (Prc)
M "Mark gets
good marks
S "Mark keeps his Scholarship"
b)
"
=
=
SEM
c) ((n)
=
"n
is
ne[meZ/m
d)
let PPCX
composite"
ab
=
,
be the
set
I<
,
netz
asban3
=
of prime numbers
(Cn)
.
ne [3 3 73 Enel
.
.
·
I E
- PE
> af
I
3.
=
T
The truth table columns under
are
the same Thus
.
,
the
PEEQ
formulas
are
and
(InQuGPTQ)
equivalent
.
3
.
b) ((PEQ) v(P r)) /(a)u(pua)
=
=
associative (PVLP) v(QUR)
idempotent <PV(QvR)
=
4. I will state
a) P (Q
=
=
P = (v)
equivalent formulas at the end of the question
r)
=
-
Pv(Q = R)
=
b)(P
=
Pv(QR)
-
Q) (P R) = (PvQ)1 (PVR)
=
Distributive
F
2) (PQ) = R
LPv(QR)
/PrQ) vR
=
De Morgan's
/FPULQUR
DoubleNegati [PVLQVR
E
~
Associative LPr(QuR)
d)P = (QR)
T
=
Pv(QR)
P= (Q = R) and (PrQ) => R
are
(Q)r(P=R) and PE(QXR)
equivalent
.
are
equivalent
.
.
5 a)
.
S
=
F
=
"you start studying
"you will fail the midterm
"
SF
b)
P(t) "t will want to eat it
"
=
M(X)
"X
=
Vt -P(t)n
,
2) TH)
=
K(t)
Mikey
is
+
"
(M(t) => P(m) nM(m)
took her class"
"t
"Knows what
=
she
great professor
is
"
a
Ft T(t) => k(t)
,
& ((t)
"t
=
"tis Anna"
.
A(t)
deserves credit
=
"
It ((t) => A(t)
,
e
6 a) Fals.
.
let x be me
my parents
letI be the set of people
.
let
P
I
auch P
b
.
UteUly <P(t p)11P(t PT
,
,
,
.
Since neither of my parents is the parent of
everyone
,
there does not exist
person who is the parent of everyone
a
b) True
c) True
·
.
Let
x
be me
. Let
y
be
Every person does have
Dr Chris Hall
.
a
.
parent
.
_
P(X y) Thus
,
.
,
.
Ex7y LP(X y)
,
,