IMS
(INSTITUTE OF MATHEMATICAL SCIENCES) Egg:
(b) a! W P521! 0555 n (fixed) rs; 01% u (strings)
_OAEI_OBF12%HQHQEBWE$(rod) ABET?!
qs-gataannEM'HOAqaiOBasmd)
ABm:aamBmm5i,aists
" (rod)$3aal%, G181
P cotoc 'Q cot B
Two weights P and Q are suspended from
What is wrong with the proof ?
If n2 is positive, then n is positive.
Proof: Suppose that n2 is positive. Because the
conditional statement If n is positive, then n2 is positive
is true, hence we can conclude that n is positive.
If is not positive, then n
First Order Logic
Clause form statements whose elementary
components are connected by the operation OR ()
First-order logic
Objects: cs4701, fred, ph219, emptylist
Relations/Predicates: is_Man(fred), Located(cs4701, ph219),
is_kind_of(apple, fruit)
Note:
IMS
(INSTITUTE OF MATHEMATICAL SCIENCES)
(lo) T7913 W $5 31$ (cone) 3yz22x2xy:0 $ 613
' maaa(generators)mwmagaa%lzr
=WWWEW (generator) 81,
mam. 7
"Show that the cone 3yz L'2zx23'cyzb has an '
innite set of three mutually perpendicular
generators. If %
IMS
, (INSTITUTE OF MATHEMATICAL SCIENCES)
2
(0) 9:11am x2y4xi+6:
dx3 dxz
(general solution) W I
=4WWTCIT=BEFI
Find the I general - solution 10f the equation I
3 - 2 * . ' - -
de34xgg+6dy=4. a 15
(d) WWI? W (Laplace transformation) W 113?: I n
TERI EFT W
IMST
I - (INSTITUTE OF MATHEMATICAL SCIENCES)
(d) Wl(sphere)$1twalfa, Egx2+y2 =4;
z=0l1313x+2y+2z=0wm
Whaamml
Find the equation of the sphere which passes:
through the circle x2 + y2 4; Z: 0 and is cut by =
the plane x+2y+22 = 0 in a circle of radius'3. 1
Direct/Indirect Proof
A direct proof of a conditional statement p
q is constructed when the first step is the
assumption that p is true, subsequent steps
using rules of inference, with the final step
showing q must also be true.
Indirect proof if we prov
Vacuous and Trivial Proof
Vacuous proof in p q, if we know p
is false already, the conditional
statement must be true.
Trivial proof in p q, if we know q is
already true.
E.g. P(n) is if n > 1, then n2 > n. Prove P(0) is true.
E.g. P(n) is If a and b are
Sufficient and Necessary Condition
A necessary and sufficient condition for an
elementary product to be false it contains at
least one pair of literals in which one is the
negation of the other.
A necessary and sufficient condition for an
elementary sum t
IMSM
(INSTITUTE OF MATHEMATICAL SCIENCES)
ENEB / SECTIONB
2 w w W (particular
dx . 2 .
Integral) WI I
Find a . particular integral. of
2 . -' .
d g+y=ex/2 sinx. ' ' 10
dx ' 2 , .
I
(b) avw 3,53% .212, Z=i+3j+4I,
c=4i2j-_ 6k
Notations of and (mutually excluded)
maxterms of p and q pq, pq, pq, pq
represented by 00, 10, 01, 11, or 0, 2, 1, 3
Minterms of p and q pq, pq, pq, pq
represented by 11, 01, 10, 00, or 3, 1, 2, 0
[(pq)p q] =[(pp )(qp)] q
=(qp) q = (qp)q
= q p q = p q = 1
The Use of Counterexamples
EX1. All prime numbers are odd (false)
Proof: 2 is an even number and a prime.
EX2. Every prime number can be written
as the difference of two squares, i.e.
a2 b2.
Proof: 2 cant be written as a2 b2
23
Disjunction/Conjunction Normal Form
Disjunctive normal form (DNF) a formula
which is equivalent to a given formula and
consists of a sum of elementary products
E.g. (p q) q (p q) (q q) is in
DNF.
Conjunctive normal form (CNF) - a formula
which is equivale
Proof by Contraposition
If (pq) (p+q)/2, then p q
Direct proof ? (not trivial)
Contrapositive:
If p = q, then (pq) = (p+q)/2
It follows by:
(pq) = (pp) = (p2) = p
(p+p)/2 = (p+q)/2 = p.
16
Terminology for Proof
Axioms statements we assume to be true
Proposition, Lemma, Theorem statement
that can be shown to be true.
Corollary
Conjecture
theorem that can be established
directly from a proven theorem
statement that is being proposed to
be a
Prenex Normal Forms
A formula F is called a Prenex normal form iff F is a
first order logic and is in the form of (Q1x1, , Qnxn)(M)
where every (Qixi), i=1,n is either (xi) or (xi) and
M is a formula containing no quantifiers. (Q1x1, ,
Qnxn) is called the
IMS
(INSTITUTE OF MATHEMATICAL SCIENCES)
and through a ring of weight 'W which is hanging
vertically; Show that the tension of the string is
W(l 3a) : '
'2\/_126la +8612 2 A . ' p _ '20
u ' :
F)
8 (a) f(r)$r1%rl, mmVf=Lsamn= or
T5
7 Find f(r) such that
Examples of Prenex Normal Forms
x P(x) x P(x)
x P(x) x P(x)
(Qx) F(x) G Qx (F(x) G), if G doesnt contain x
(Qx) F(x) G Qx (F(x) G), if G doesnt contain x
E.g. 1 x P(x) x Q(x)
sol: x (P(x) Q(x)
E.g. 2 x y z (P(x,z) P(y,z) uQ(x,y,u)
sol: x y z u (P(x,z)
IMS
(INSTITUTE OF MATHEMATICAL SCIENCES)
. degree at most 2, then nd the matrix
representation of T: M2 (R) > P2 (x), such,
' that -_ THCl bD=a+c+(aId)x+(b+c)x2,:
. c- d
with respect to the standard bases of M2 (R)
and P2 (x) Further nd the null space of
Principal Disjunctive Normal Form
Minterms of p and q pq, pq, pq, pq
each variable occurs either negated or
nonnegated but not both occur together in the
conjunction.
Principal disjunctive normal form for a given
formula, an equivalent formula consisting
Proof by Contradiction
2 is irrational
Suppose 2 is rational. Then 2 = p/q,
such that p, q have no common factors.
Squaring and transposing,
p2 = 2q2 (even number)
So, p is even (if x2 is even, then x is even)
that is, p = 2x for some integer x
hence, 4x2
IMSM
(INSTITUTE OF MATHEMATICAL SCIENCES)
Using elementary row_ operations, nd the
_ "condition that the linear equations
x2y+z"=a_ :
2x+7y3z=b .
.'3x+5y22'=c 1 _ . '
. I. have a solution. . 7 , 7
. (ii)2r
' , M#cfw_(x,y,z)IX+y-Z=0
~ _ . ; W2=cfw_(x,y,
IMS
.f . (INSTITUTE OF MATHEMATICAL SCIENCES)
(ii) 111% W % Emil (Hermitian) MEL5' a; Fi
TWWWamasl
Prove that eigenvalues of a Hermitian matrix 1
are all real. .' - .' 8
. l
I - . l , o
.4 . 1
(c) 4% Ianm (bases)! Icfw_1 Ix, x(1 x), xg1 + x) 12-4
cfw_1,
Existence Proofs (1/2)
E.g.1 There exists (distinct) integers x,y,z
satisfying x2+y2 = z2
Proof: x = 3, y = 4, z = 5. (by constructive
existence proof)
E.g.2 There is a positive integer that can be
written as the sum of cubes of positive integers
in two d
Proof by Equivalence
n is even iff n2 is even
Proof (by equivalence)
Let P be n is even, Q be n2 is even
P and Q are equivalence can be proven by
P Q and Q P
24