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Suppose that S is the set of students and C the set of courses, and G be the predicate on the set of courses such that G(y) = “y is a graduate course”. Let U and R be predicates on the set of students such that U(x)= “x is an undergraduate student” and R(x) ”x is a graduate student”. Finally, let T be the preciate on S x C such that T(x,y)=“x is taking y.”
Translate into predicate logic the proposition: “There is an undergraduate who is taking only graduate courses.”
There is a student x such thet x is on Tsees(Uhc)n (Hye((Tbc,g)→ Gly))))
for every course y, if x is taking y, then y is a graduate course .
undergraduate
stye((Uhc) ^ (The,g)→ Glyn) Translate the following into good English:
thees(r UCH n VyellGlylsThe,y)))
c student x
not o n fees. so is
urekrgocdacfe
There is no graduate student simultaneously toking every gradate course .
for every course y i.e .
t .t l The point of logic is to clarify proofs .
In meth, we want to prove theorems , which means establishing
propositions
extent , their form
techniques
for proving
propositions
importantly
constantly
Hx(Phd) and
are actually true .
eg f” If a number hes a reseating decimal expansion ,
” retinal .
ke KY ( 8 – 2kt
” number larger then 1 is divisible
fu x> I → Fp x is divisible by p.
completely
theALPIN ) , we take
a parameter a
representing
element a t A then 2 is
31 somehow odd .
G- Every prime greeter
If P is the set of primes,
we’re saying
where Else): xiseven :
2k ) . a:3:’s { Letpepb- e anyprime Now we need to prove g>2→ a Elp).
know this is automatically true whenever p > 2 is
faceP x> TECH
w e re so ,
need to address
then p> 2.
be even because
> 2 Suppose p .
Then p k then
would not be prine (
p , p noting
tht k> 1 since
So, how do we prove an implication of → r ?
As we saw above , the most straightforward approach is
or modes ponders tf n is even, then so
q is arbitrary ,
sqelltout assumptions),
r . thendo meth to. . .
Notice the formel weening !
‘) n is even → n is even.
Let me# be any integer
Elm) → Elm’) by direct proof. milk for sore KE E We went to
Squaring both
assure Echl
m2 is even ,
Thet hens .
21 for some
I ‘d get in ?
be Kraig m=m2
Thus n: 2(2b) so we can set 1=262. ,
we finished this by proving The# (n’=2l).
existentially
quantified s t o l e n ‘ t Ket 1=262
tf f question:
works . Note Participation: Edgar, Esai
lot different
ae Or be or } nhopnossibitieso
, 10×01–101.101=0 –
3/3:Moreonproofs_
Last time , we explained how to
) I let a be an arbitrary element ofA , prove Ifad)
oftefgwed.ane.%ee.it.lt
Prove tht not every prime number is odd
them let Mint ” n
2 is not For , – .
odd M(2) n 70121 , n (M121 → 012)) , so ” ( → Ocn) )
Fnl’ (Mint→ Oln)) an.
= theNll- Mnl 01mi)
Most often implicational → RH
We introduced direct proof for ) o n implication : pretend antecedent is true ,
:*:*. notation .
Two further ways to prove implications .
derive the
✓:3 nai ‘E’IIF’t:c :&
Proof by contrapositive We know
=’ar→ soheooh
Sha thet if EH )
be false n,
of 7g → is
indeed p e a r
is false, p must
III.If ne# and I’ is even
tf ‘ is even
there’s # St
We need to use this to show well for som l . uhh . ( hndrrect proof : ) Assam n Elm ) show r Elm’ ) n Elm) means
Indirect proof is great for
– KEE ) 2k-11 for some . hewn
m . This means m –
m2:(2k¥11 is odd as well
4k’t4k -11 RT
212k ‘t2k ) -11 even
(grrrl . 0cal ore
instance , ?!
p → tape#(Olathe (Oldr 01611)
ha to prove
A and b are both even do
Direct proof – – –
✓ 01611 , i.e
n -0161, i.e.
So a– 2k , 6=21
ath Ukell B
Proofbycantradiciion
( Manta men ) ) o r
prove I infinitely many primes
→ET} Mlu’ ”
clever fornelieetim :
the Nl (Mini → Time
P Fore P ( pl q
How to prove ? Straightformally would be to find a q > p for a n
arbitrary prime p No way ! Trick : Instead of s
7-pep HqEP pep
wast be false too ?
Then the set of fries world be finite g P:{ ,
is s a n e 9,32 93,6
Cen w e show There hast be
a bigger … .
grin after
by any of p.
, pie yn +1
qz.no a more
divisible by some
other prime q # {a , – niece } .
ill the serines Contradictor ! Sols
: ” must be false .
. primes ”
There is DZ
So t proven
serine not in the set of
x.ge#scehlhetx2–4yt2
262=26+1 . The is odd, contradiction ! B
‘ is even and
is even . 462=21261-11
Leta soa= ,
Prostheses you can prove fret ( Pla) ) by proving
theeA(Q,lxlv V Qulxl)→PIN, ifViceA(Q.Coolv. . . also)listee … .
FEI V-lx.gl EIR ‘ (
lxtylelxltlyl
4 x?0 erdyzo, or 31 or xeo
31ye0,or3xzo YEO, 8 oralyzo
lxtyl-xty.lk/–x ,
4 Hk- x Iyl- – Y ,
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