===================-:::2,. : :J:!ntstt.: iff l!.JI.f!i:!

i/iiit:r:-·S:-====--:=-===-=====-===-==-:::::-::::::~
State whi ch of the follo win g are stat eme
white.
1. All cro ws are blac k and all swans are
2. Ram a is running.
rea che d.
3. Arise, awa ke and stop not till the goal is
4. Wh at a bea utifu l piec e of liter atur e !
5. A Cretan says, "All Cretans are liars."
6. Silv er is white and iron is blac k.
7. If 4 > 3 and 3 > 2 then 4 > 2.
8. Are you a fool ?
9. If 3 > 7 and 9 > 11 then 8 > 5.
nts, othe rs are not./
[ Ans. : (1 ), (2), (6), (7), (9) are stat eme
 If Ais a true st at em en
t an d X is a false st
atement determine
1. . the truth value of ea
ch of the
statements.
follow1n9 · . X (ii) N A V X (iii) AA N-X
A" "'
(i) "'
. (iV) "' (A" X) (v) rv (A v X ) (vi) {rv (A
N
X ) /\ N A }
I\ N

[ Ans. : (i) F, (Ii) F,

If A and B ar e true st at (iii) T, (iv) T, (v) F,
- - ... (vi) T ]
em en ts and X is a fa
2. lse statemerit, determ
ine the truth value of
following. the
(i) A V ( B " X )
(ii) (A A X ) V ( B " X )
(iii)"' XA("' Av,.., B
) (iv) [(A" X ) v "' BJ
· (v) "' "' [ ,.., (A A ;.., X) v [( B v, .., X)",.., A]
v ,.., A ] " "' B ]
[ Ans. : (i) T, (ii) F, (iii)
F, (iv) F, (v) T ]
 ent and Xis a false statem en t ~eterrni ne the truth value of the following,
1. 11 Ais atru estatem
(i)i A-:J (A-:J X) (ii) (X :J X) ::> A

(iii) (A ::i X) ::i A (iv) A:J (Xv A)

(vi) [(A" X b NA ] :J X
I

(v) (A::i X) ::i (N X::i NA)

(vii} (Av X)::i ( A" X)
N N N (viii) [(X :J A) :J A] :J (A v N X)
(iii) T, (iv) T, (v) T, (vi) F, (vii) F,
(viii) T]
l Ans.: (i) F, (ii) T,

F - -- - -
 If A an·d B are t ru e statement s and
· X is a f'a\se statement
., determ,ne '\'.he truth ,,a\ue 01 th e
2. . g
fol lOWln .
(i} A :::, ("" B ::) X )
(iii) (A :J B) :J ("' X :J "' B) (iv) l(/\ " B) ::) X 1 ::) l A::) (B :) X )1
(v) (NA::) X) V (B :) rv X) (vi) (A :) rv X) " ( B:) X )
_(vii) ~ (A v X) ::) B (viii) r:v (A v rv B) :) \_8 " ~ ( X:) A)1
[ Ans. : (i) T, (ii) T, (iii) F, {iv) T, {v) T, (vi) f , (vii) l", l~ll\) l" 1
 o~V - - ;:r "'
·5Cr
1
0 . state the converse, inverse and the contrapo •t·
pie 6 · si ive of the foll ·
~~~rt1 . . cold then he wears a jacket. owing :
. If it is .
(1) ·nteger is multiple of two then it is even
,. If an I • (M
(11) . it is cold and q: he wears a jacket. .u. 2013)
') 1..et P ·
I : (1 have P :J q
so. en we
fh e is q => P
onvers
11s c a i·acket then it is cold.
wears
If he
·nverse
.
is
~ ~p :J q
1
fhe_ t cold then he does not wear a jacket.
If it IS no .
ntrapositive 1s ~ ~q :J P
115
co not wear a jacket then it is not cold.
If he doe 5 . . .
. n integer is multiple of two, q : 1t 1s even.
··,) 1..et p . a
(I ve P :J q
Then we ha
11s converse is .q .:J P .
. teger is even then 1t 1s a multiple of two.
If an in
. ~p :J ~q
Its inverse is
If an integer is not a multiple of two then it is not even.
Its contrapositive is ~ q :J ~ P
If an integer is not even then it is not a multiple of two.

==::::=================~i ~ig~~!~it I ~~-====================

Write the converse, inverse and contrapositive statements of the following.
(i) If you are honest then you will succeed.
(ii) If you work hard then you will be rich.
(iii) If you are healthy then you can be wealthy. [ Ans. : Left to you ]
1 . lt A ts a t rue statement and Xis a false statement determ in e the truth va lue of
(i} ( '"" A= X) /\ A (ii) (A I\ rv X) = A
(i i i) ( A ::J X ) = ( X ::) A) (iv) [(A/\ X) ::) A ] = ( rv A V X)
( v ) ~ (A= X ) A A (vi) rv (A= X) V rv rv A
[ Ans. : (i) T, (ii) T, (iii) F, (iv) F, (v) T, (v i) T]
 If A and B are true st atements and X is a false statement determine the truth va\ue of
z. (i) (A::) B) =(B:::, X) (ii) (A" X):::, (B =X)
(iii) (B =="' X) "(Av X) (iv) A::) (B = IV X)
{v) ("'Av N B) = (A :J X) {vi) (A:) 8) = ( Av B) IV

(vii) "" (A " B) = N A v "' B (viii) IV (Av B) = IV A /\ IV _ B

{ix) (A :J X) = (A :J "' B) (x) IV (A= 8) " X
{xi) (A:) B) ~[(A:)"' a)" X1 (xii) (A:) a)= llV B:) A 1 IV

t Ans.: (i) F, (ii) T, (iii) T,. (iv) T, (v) T, (vi) T, (vii) T, (viii) T, {ix) 1", (x) F, (xi) F, (xii) T 1
 !
·.·>.·.•,•.·.·.·>.·.·.·>.·.•.·:•:·:•.·.•,•.·.·.·.·.·.•.•:•:•:·:·:•:•.·,·.·.·¥·'
=-==
===::=:::::::=:::::=::::::::=:::::::=:::::::::=========-==~=_:=:~~::/~X
·.·.·:.·.·.·:.·.€~ . . .·.·.·.·.·.·.·.·:.i ·i.f·.·.\V
·.·.·.·R.·.·..~!$.~ ·.·.·.·.i·~
.·.·.\/_~=
·.·.·. ===-===::::::::::::::~
. Construct truth-tables to show the truth-values of the following statements.
1
(i) ( p I\ q) :) p, (ii) (p V Q) ::J Q, (iii) (p /\ p) ::] p,
(iv) (p v q) /\ N r (v) (p A N q) v r (vi) (p :) N q) = r
(vii) (rv p :J r) =q (viii) N (p I\ q) /\ (p = q) (ix) (p /\ q) -:J (p v q)
(x) p:> (rv P:J q) (xi) (Np V q):) N r.
2. Prove the following equivalences by truth-tables :
(i) ( p:) q) : P V q N (ii) N ( p V q) = p /\ q N N

(iii) Npt,. q) : p V q
( N (iv) ( p I\ q) = ( q I\ p )
N

3. Using suitable symbols for the component statements symbolise the following :
(i) Milind and Sunil will go if and only if either Tushar or Yogesh won't go.
(ii) Democracy can survive if and only if people are vigilant of their rights and responsible
to their duties.
(iii) lt is not true that a quadrilateral is a rectangle if and only if its opposites sides are
parallel.
(iv) If Sunil is poor he will not go abroad and conversely.
(v) A triangle is equilateral exactly if its all sides are equal.
[ Ans.: (i) (M /\ S) =(T " Y), (ii) 0= (V /\ R), (iii) N (0 = P), (iv) S = A, (v) S = A]
 •[1 , (p :J N q) == .(

i 4. l P ,,, q ) :J ( P v q )
lences by tru th -ta bl es :
tBl Prove the following equi va
,v ( p V q) =:
rv p /\ rv
q 2.
j . ( p ::J Q ) ==- N p ==.
q 4. ( p :: q) :: ( q: ) ) q
3. p (\ q) ==. N p V N

q) S, ( ,v q : NI \ ( q::, p )
N (

) p q) 6. ( p::)
s, ( q cap) a,; ( p A q V ( N I\ N

l rv
l 8. ( p = q ) ::) [ (,v Pv q) P l\(
1. l ( p " q ) ::i r l =[ P ::i ( q ::i r)-
V ( q /\ f) = ( pv q) " (
. p q V p))
10. ( p p ) = p 11
9, ( p /\ p) = p V

r) 13 . p = [ p v ( p :) q ) J. P v r)
i 2. p1d q v r ) =( p " q ) v (
P v
 " - •
Ashokn Wll1S lfl U y ,111 1u llll \.• , , , '--- ---

====~ < F
f- -====================~
===-========
===
EXE
~ ~
~_ l~ £ - VII l
following sta lcrn onts in symbolic form
.
I

1. Usr ng suggested symbols write th e

stud ies San skrit (A) and eith er Bro wn stud ies English (B) or Charlie doe
(r) Ashoh.a s '1()1
study French ( C).
nch (
is not true tha t Ash oka doe s not stu dy Sanskrit (A) and Charli e studies Fre
(u) It wn C)_
er Cha rlie stud ies Fre nch ( C) and Ash oka does not study Sanskrit (A) or Bro
(Iii) Eith doos
Hindi (0) .
not study English (B) and Din kar studies
a
na get s gold me dal ( C) and Britai~ does ~ot win the final game (B) or if Brit 1
(iv) Either Chi
not w,n the silver medal (A) .
~

win s the fina l gam e the n Ala ska doe s

cas e tha t bot h Ala ska win s the silver medal (A) and Britain wins the fi
(v) It Is not the ~~I
then eith er Brit ain doe s not win the final game or China gets the gold med
game (8)
fina l gam e (B) the n Ala ska win s the silver medal (A) implies that Chin ~
(vi) If Britain wins the
does not get the gold medal ( C).
go.
only if Tushar (7) and Yogesh ( Y) will
(vii) Milind (M) and Sunil (S) will go if and
osite sides are
is a rectangle (0) if and only if its opp
(viii) It is not true that a quadrilateral
parallel (P). ~
C) (ii) N (~A " C)
~
(iii) (CI \ A) V ( B AD)
f Ans (i) A" (B
. : v N

:J (~ Bv C) (vi) (B :J A)
:J~ C
(iv) (C " N 8) v (8 :J NA) (v) ~(At-. B)
(vii) (M " S) =(T t-. Y) (viii) N (0= P). ]
statements in words.
2. Write the negatives of the following
(i) Rama is tall but weak.
(ii) Either Xis elected or Y is selected.
(iii) Rama swims if water is worm.
ill.
(iv) Rama is absent if and only if he is
uring first class implies that the student is intelligent and hard working.
(v) Sec
2 2
(vi) If x = y (A) then X= y(B).
then the side a= the side b ( q).
(vii) In a triangle ABC, if LA = LB (p),
Kant were not
o were both not artists or Hegel and
(viii) Eith er Leonardo da Vinci and Picass
both philosophers.
isosceles
+ 2 = 6 (S) althoug h 5 + 1 = 4 (F) or squ are is a circle ( C) but any triangle is a
(ix) 3
(I) .
ficient condition for healthy life (H) .
(x) Good Liver is (G) necessary and suf
s.: (i)~ (T " ~ = T v W. ~ Rama is not tall or Rama is not weak.
[An N

(ii) rv (X v Y) =rv X" rv Y. Xis elected and Y is not selected.

(iii) ~ (W:J S) = ~ (~ Wv S) = ~~ Wt-

. ~ S =Wt-. NS
swim.
Either water is warm or Ram does not
 Logi c

A ::> I) /\. (/ ::> A) ) = ~ [ (~ A V / ) /\ ( ~ I

V A) l -

(iv) ,., [ ( ~ ~
= ( ~ A /\ ~ I) V ( rv , /\ ~- ~(~ ~(~/
A V ' ) V V A)

A) == (A /\ "' I ) v (I " ~ A )
arns · a b se nt and h e ·1s not ·1
1s I I or Ram . .
s is ill and h .
·11,er R H)) =~ ~F /\ ~(I e 1sno ta.b sent .
f1 JA H) =~ (~F l\ (1 1\ /\ /-f) =:F /\ ( rv f v rv
,-,( f ::> . . H)
er he is not . t .
(v) d nt sec ures first clas s and eith in elligent or he ·
stU e is not hard working .
~(~ A v B) = ~~ A I\ ~ B = A " ~ 8 . . 2
A 8) = .. X - y 2 d
(,,i)
,.,(A ::>
~(~
::> q) == ~~
p v q) = p I\~ q = p /\ ~ q - an x 1:. y.

(1,ii) "' (P ·ang le ABC L A = L

Ban d the side a * the side b.
In a tn
rvP )v( ~H v~ K)]
.. r-J [("' LI\ NP )VN (HI \K) ] =~ [(~ L/\
(Viii) = ~ (~ L I\ ~ P) /\ ( ~~ H V ~ K)
= (~~ L V ~ P) /\ (~~ H. ~~ K)
= (L I\ P) I\ (HI \ K)
t were both Ph'II osophers .
rdo da Vinc i or Pica sso was artis t and Hegel and Kan
Leona
('J [(S /\ F) V (CI \ I)] =~ (S /\ F) /\ ~ (C /\I) =(~
~
s V F) /\ (~C V ~I)
. .
4 d le or any triangle is not isosceles.
~x)
+ 2 -:t 6 or 5 + 1 #- an a squ are 1s not a circ
x)
3
G :J H) " ( H ::J G) ] =
('J [ (
~(
G ::J H) v ~(
H ::> G)
( =[~ (~ Gv H) ]v[ ~(~ Hv G) ]
=(~ ~ G/\ rv H)v (~~ H" ~ G)
~
= ( G /\ H) V ( H I\ G) ~
life is healthy and liver is not good.
Either liver is goo d and life is not hea lthy or
- ===~=== ==- -•:\ :i x'.t '.~ e x·::·:·: : .: ::0::· · · :;:. :;: :-.:·:
====-======-======-3::::=:::::-==-==d:::s
· ... .·.·.·.·.·.-:•:-:-:•:-<-:•:.~ .~ :.:-:-·>:Y..l Il· -==-:::::==--=:::::::=.::::===.::=:::::===-:::::::::::::::
construct truth tables to determ· ... .......·.· ..-:::::.:::::..::
contradictory or contingent. me Whether the following sIa1ements f
orms are tautologous '
(A) 1. p -:J N p 2. p :) p
5. p:J ( p /\ p) ;• p -::; ( p -::; p)
6. p :J ( p V p) 4. (p -::, p)-_:, p
8. ( p :J p) /\ ( p:) p) N N . ( p J p) /\ ( p :) P)
9 (p ) N N

p /\ ( p V p) • 10 ( :) N

[ Ans. : 1. Contingent 2. Tautology 3. Tautology . N p -::; P) " (Np)

4. Continge nt
5. Tautology 6 T
. autology 7. Contradictory 8. Tautology
9. Contradictory 10. Contrad ictory l
(B) 1. p :J ( p " q)
2. ( p /\ q) :) p
3, P V N ( P /\ q)
4, ( N p /\ q) /\ ( q:) p)
5. [( p·:J q) /\ N q] :) p
6, ( p :) q) /\ (N q:) N p)
7, [( P V q) /\ N P] :) q p] :) q
8. [( p:) q) t\

9, ( P :J Q) :J ( N p V Q) 10, [rv(p/\ q)]:)[r vpvrvq ]

11. [~(pv q)]-:J [~p/\r vq) 12. ( p :) q) V ( q ::J p)
[ Ans. : 1. Contingent 2. Tautololy 3. Tautology 4. Contrad ictory
5. Contingent 6. Contingent 7. Tautology 8. Tautology
9. Tautology 10. Tautology 11. Tautology 12. Tautology ]

(C) 1. [p-:J(q -:J r)] -:J[( p " q) ::J r] 2. [(p ::J q) /\ (q ::J r)] ::J (p ::Jr)
3. [ p /\ ( Q V f)] :) [( p I\ q) V (p t\ ( )] 4. [ p v ( q I\ r)] ::J [( p v q) /\ ( p v r )]
5. [p -:J (q -:J r)] :J [(p :J q) :J r] 6. [ p ::J (q ::J r )] ::J [ q ::J ( p ::J r) ]

[Ans.: 1. Tautology 2. Tautology 3. Tautology

4. Tautology 5. Contingent 6. Tautology. ]

 1. Write the negation of the following sta
tements.
(i) 4 is even and - 5 is negative.
(ii) 4 is even or - 5 is negative.
[ Ans. : Sy~bolise the given statement and negate it usi
ng De Morgans' laws
(i) 4 is not even or - 5 is not negative.
(ii) 4 is not even and - 5 is not negative.
]
2. Write ~the negation of the following
statements.
(i) If 2 is prime then 3 is even.
(ii) A triangle is equiangular if and only
if it is equilateral.
[ Ans. : Use ( p ~ q) = rv p V q.
(i) 2 is prime and 3 is not even.
(ii) A triangle is equiangular and it is not equ
equiangular. ] ilateral or a triangle is equilateral and it is
nol
 3. Using the laws of log_ic $imp\ify.
(i) ( .p A Q) V p . . (ii) p A ( p A Q)
. (iii) p V ~~ p A.· q) · (iv) .( p V "' ·Q.) A~ ("p A Q)
· [ Ans. : (i) p , (ii) p " q, (iii) p v _· q, (iv) p. 1 -- -·.
 rmin e the truth value s of the folio .-
1. If the univ erse of disc ours e is A= {1, 2, 3, 4, 5, 6} dete Wing
prop ositi ons
(i) (3x) (x + 2 < 4) {ii) (Vx) (x + 5 < 1 O)
2 2
(iii) (3x) (x < 4) {iv) (Vx) (x - 3x + 2 = 0)
[ Ans. : {i) True , {ii) False , (iii) True, (iv) False.]
value s of the followir_,g propositions
2. If univ erse of disco urse is {1, 2}, dete rmin e the truth
3 2
2
·(i) (3x) (x = 1) (ii) (3x) (x - x + x = 1)
5 4
2
_ (iii) (Vx) (x > ·2 ) (iv) (Vx) (x + x = 12)
[ Ans. : (i) True , (ii) True, (iii) False, (iv) False.]
~ i l~i~l~i t::iJ ===============
Write the negatives of the following propositions.
J~:>

(I) _ for all positive integers we have x divisible by 7.

1
There is a person whose height is 8 feet.
2, 2
3, for all integers x, x > 0.

4•
For all real numbers x, if x > 5 then x 2 > 1o.
2
s. For all real numbers x, if x + x + 1 = o then x > 2 or x < _ 3.
6. For all real numbers x, Y, z, if x- y is even and y- z is even then x - z is even.
[ Ans. : (1) there is a positive integer x such that x is not divisible 7 .
(2) No person has height 8 feet.
(3) There is an integer such that x 2 ;/> o.
(4) There is a real number x such that x > 5 and x 2 -:;. 1o.
(5) There exists x such that x + x + 1 = O and x ;/> 2 and x 1- - 3.
2

(6) There exist real numbers x, y, z such that (x- y) is not even or y- z is not even or
(x- z) is not even.]
(II) Negate each of the following propositions.
1. (Vx) (Vy) P (x, y) 2. (Vx) (3y) P (x, y)
3. (3x) (Vy) P (x. y) 4. (3x) (3y) P (x, y)
[Ans.: Remember~ (Vx) P (x, y) is (3x) ~ P (x, y) and~ (3x) P (x, y) is (Vx) ~ P (x. y) . Hence,
(1) (3x) (3y) [~ P (x. y)). (2) (3x) (Vy) ["' P (x, y)],
(3) (Vx) (3y) [~ P(x, y)], (4) (Vx) (Vy)["' P(x, y)].]
 I ,, I , , """' -
• .. - •- ... - ,. -, -· ·,
__
- ,,,, .
... ,.. ,., .., ~~ m
7. "' .~ ':- :0 :,~
=========::::::::==============
=~ t i ~jpjjj:f ii~ _
· propositions.
. se of discours
1. If the un,ver e is {1, 2, 3, 4,
5, 6, 7, 8} ' d e te
rm in e th e truth
I\
following \ i
(i) (v'x) (3 y) P (x, + 0) va.1\J~
Y< 1 (Ii) (Vx) (Vy) P (x
+ y < 10) ()! \h
~
1
2. Dete rmin . the truth value o [ A n s. : (i )
e · f th e fo llo w in g p ro T ru e ('' ) \
p o si ti o n s if the
LJ : {i , 2, 3, 4} . u n iv e rs e of 'd' II Fa.1
(i) (3 x) ( y) (x 2 < 1sco1Jrss~ .11
3 y + 3) (ii) (Vx) (3 y) (x
2 2 . e Ii I
(iii) ('v'x) ('v'y) (x 2 + y < 20) I
+ / < 20) (iv) (3x) (Vy) (x 2 2
+ Y < 10)
. ( A ns. : (i ) T ru e, (i i)
3. If the universe of d1~course . . T ru e , (i ii) False
th e truth value of th 1s ~~ e se t of in te ge rs an d . 2 .
, (iv) F I
e tollowmg propos P (x ) . x > x, 2
0 (x) . x = x, deterrn al se \
1t1ons. \ne
(i) (\7'x) [N p (x))
(ii) (3x) [ Q (x)]
(iv) (3x) ( P (x) v Q (i ii) (3 x) [ P (x ) /\
(x) l (v) (Vx) [ P (x) /\ Q (x )]
0 (x )] (v i) (V x) [ P (x )
[ Ans. : (i) False, (i v Q (x )] \
4. If the universe i) T rue, (i ii) F al se ,
of discourse is th (i v) T ru e , (v ) F al se ,
propositions in sym e se t o f re al n u (vi) True.
bols and determin
e its truth value.
m b e rs , re w ri te
e a ch o f th e follo 1
(i) The product of w in g
any two real nu m
be rs x an d y is po
(ii) There are real si tiv e.
numbers x, y such
x = 2y.
(iii) For each real
number x there is
a real nu m be r y,
(iv) For each real su ch th a t x x y =
number x there is
a real nu m be r y x.
[ A ns . : (i) (\7'x) su ch x + y = x.
(\7'y) (x y > 0) ; Fals
e, (i i) (3
(ii i) (Vx) (3y) (x x x) (3 y) (x = 2y ) ;
y = x ) ; True, T ru e,
5: _If the (i v) (V x) (3 y) (x + y = x
prop osrtr ons. un iverse of discourse is th e set ) ; T ru e ]
of in te ge rs , fin d
th e tr ut h va lu e
(i) ('v'X) (x 2 o f th e following
> 0)
(iii) (3 x ) (3y ) (x +
y = 13)
(ii) ~ (3x) (x 2 = 3)
(iv) (Vx) (Vy) (x +
y = y + x)
[ Ans. : (i ) T ru e, (i
i) T ru e, (iii) Tru
e, (i v) True. ]

- - -- -
 (x ): x?: y,
rs , p (x , y) : xi s a multiple of y ; Q
6
llotwof intege
un iv er se of
lu e di
of sc
eaouch e is
rsof e efose
thth ing.
e th e
rrn·.1nIf the tr ut h va
x) (iii) (\ix) P (x, 2)
(ii) (Vx) P (21,
dele (i) (3 x) P (21, x ) i) (\ix) (3 y) Q (x, ..Y )
v Q (x, 6) ] (v
(v ) (3 x) [ P (x, 2)
~) (3 x) P (x
(ivii) (3 x) [ Q (x, 5)
(v
, 3)::, P (x , 5) ]
True. l
(iv) Fa lse, (v ) Tr ue, (vi) True, (vu)
\ ) False, (iii) False,
[ A n s. : (i) Tr ue , (ii
 . . • ''Y J

================~ Mii9Jit{At4~~tJ# ~~~~!i~i }------==-~

1. Using the followi ng symbo ls :
p : I will study discre te structu res.
q : I will go to a movie .
r : r am in good mood.
write the foflowi ng statem ents in symbo lic form.
(i) If I am in a good mood then I will go to a movie .
(ii) I will not go to a movie and I will study discret e structu re.
(iii) I will go to a movie only if I will not study discret e structu res.
(
(iv) I will not study discre te structu res then I am not in good mood. M.U. 1996)
[ Ans. : (i) r-:::) q, (ii) rv q /\ p, (iii) q-::) IV p , (iv) ~ p:) "' r.l
2. Transl ate into symbo lic form the followi ng :
(i) If I play foot-ba ll then I canno t study.
(ii) Either I play foot-ba ll or I pass.
(M.U. 1998)
(iii) I play foot-ba ll therefo re I pass.
P, (iii) F-:J P.]
(Ans. : F: I play foot-ba ll. P: I pass, S: I canno t study. (i) F-:::J S, (ii) Fv
3. Using the followi ng statem ents :
p : I am bored.
q : I am waiting for an hour.
r : Bus has not arrived .
translat e the following into English langua ge.
(M.U. 1994)
(i) ( q V r) ::::> p, (ii)
IV q -::)
IV p, (iii) ( q-:::) p > V ( r-:::) p )
1Ans. : (i} If I am waiting for an hour or if bus has not arrived then I am bored.
(ii,) If I am not waiting for an hour then I am not bored.
(iii) I am waiting for an hour then I am bored or if bus has not arrived
then I am bored. 1
4 • Let P denote "Raju is rich", q d~note "Raju is happy" . Write each of the followi ng in symbolic
form :-
(i) Raju is not rich but happy.
. is neither rich nor happy.
( ii) R aJu
(iii) Raju is either rich or unhapp y.
(M.U. 1996)
(iv) Raju is not rich or else he is rich and unhap py.
[ Ans. : (i) ~ p A q, (ii) rv p A q, (iii) p
IV V rv Q, (iV) rv p V ( p /\ rv q ). ]
 Logic
5· Let p b e ..
H o Is to ll" ond q I "t--l o I., ha nd
bolic form . ~ so m e" W ·t
5y111
(I) H e Is toll an
JO
· °
n Ouch of th e
following statem ·
ents in
d ho nd sorno .
(II) H e Is toll bu
t no t ha nd so m e
(iii) It Is fats o th .
at ho Is not toll
or ha nd so m e .
(iV) H e is ne ith er
ta ll no r t,o nd so
me .
(v) He is ta ll, or
he Is no t ta ll an
d ha nd so m e .
(vi) It is no t tru e
th at he ts no t ta
ll or not ha nd so
me .
{An S. ·• (i) p " q, (ii) p t\ q, (Iii)
~ ~(~ p v q) (Iv)
(v )p v (~ p A q) , ~ p t\ ~ q
, (v l) ~ (~ p v ~
Write the followin q) . ]
g statements in '
6 symbolic form .
· (I) India will w
in the word-cup ,
if the fielding Impr
oves .
(ii) If I am not in good mood an
d I am not busy
the
(iii) If you know object oriented pr 1 .
ogramming and if n WI 11 go to a movie .
·v) y k
(1 I•will score good marks in the ·
examination If an ou now oracle then yo
l Ans-: (i) Let I .• lnd_1•a • •
onl If u will get a job
w in s world -cup. Y 1 study hard
F : Fielding improv . (M .U. 19 96 )·
=
(ii) M I am m go
=
od mood, B I am
es . F =, I .
(~ M /\ N B ) ::> G busy, G = I will go
to a movie .
(iii) p : You kn ow
ob je ct oriented pr
ogramming. q =
r : You ge t a job. ( You know oracle
.
p /\ q ) ::> r.
(iv) p : I will scor
e go od m ar ks in
the examination.
q: I study hard.
p =q ]
Let p denote the st
7· at em en t "t he food
the statement "the is good", q denote
rating is three-st the statement "th
10enote ar. e service is good"
Write the following
st at em en ts in sy '
mbolic form.
(i) Either food is
go od or se rv ic e
is good or both.
(ii) Either food is
go od or se rv ic e
is go od bu t no t bo
(iii) The food is go th.
od w hi le the se rv
ic e is no t good.
(iv) It is not the ca
se th at po th th e
food is go od an
(v) If both food and d th e rating is th
service are good th ree-start.
(vi) It is not true th
en th e rating is three-star.
at a th re e st ar ra
ting al w ay s m ea
ns go od food an
d good service.
lAns. : (i) p v q, (ii) ( p /\ (M.U. 1993)
q ) v ( N p /\ q), (ii
N
i) p " q, (iv)
(v)( p " q ) ::> r, (v N
p /\ r),
8. Write English se nt
i) ~[
r :J ( p /\ q ) ]
N (

en ce s co rr es po
nd in g to ea ch of
(i) (Vx)(3y) R (x, y) the following.
(ii) (3 x) (v 'y ) R (x
(iii) N (3 x) P (x ) , y)

\Ans.: (i) For all x, th

(iv) ~
(v' x) P (x )
(M.U. 2005)
ere exists a y such
that x and y are re
(ii) There exists an lated.
x fo r all y such that x
and y are related.
(iii) There does no
t exist any x such
that P (x) is true.
(iv) P(x) is not true
for all x. ]
 9. Given the truth values of P and Oas T and those of Rand S
as P, find ti..
statement. . '~t ~11
o
r P I\ ( I\ R ) J v N [ ( P v v ( R v 1 o) s) rtJt h"
qj~
: o 1

(M ~c1
( Hint : Putting the given truth values. 11

.lJ. 199 9 \
Expression = [TAT A F] v N [ (T v T) v (F v F) J I(~ I
11;
= [TAF ]vN[ TvF] 1

= F V ( N T) = F V F = F. )
q
1o. Prove by the truth table : [ ( p v q) A ( p v "" q) v q] == P v
[Ans.: (M.u
. <o
p q pvq tvq pvtvq (pvq) t..(pv tv q) (pvq) l\(pv" '
~;)
q) vq
T T T F T T
T
T F T T T T T
F T T F F F T
F F F T T F F

Third and the last columns prove the equivalence. ]

11. Prove by truth table : ( p v q) =( p A "" q) v ( q I\ "" p) v ( p I\ q)
[Ans.:
p q r.1q pt..r.1q Np qt..tv p p Aq (pt..N q) v
(q t.. tvp)
(Pt.."" q)v
(qt..tvp)v(pt..q)
--
Pvq

T T F F F F T F T T
-
T F T T F F F T T T
F T F F T T F T T T
F F T F T F F F F F

The last two columns prove the required equivalence. )

12. Construct the truth table : "" [ p I\ ( p v q) ] N (M.U. 199~)
[Ans.:

p q r.1q P V NQ p 11 (p VNQ) N [p A(p y N Q)]

T T F T T F
T F T T T F
F T F F F T
F F T T F T

(M.U. 2000)
13. Verify whether p v N ( p I\ q) is a tautology.
[Ans. :
p q p t.. q N(p 11 q) p v rv (p 11 q)

T T T F T
T F F T T
F T F T T
F F F T T
,~'•' Logi c
. ct the truth table for the sl ntement ( P )
ons\1U , . :l q ;::: ("1 Q '.J N •

1A, C . P ). Is it a tautology ?
\ ~ns, ·
I
r---i I NP (M.U. 1996)

I---
p q p =i q N Q I

"' q -) "' f)
(p - >q ) -=> rv ( q :::> Np)
T T T F F T
T F F F T T
F
F T T T F T
T
F F T T T T
I T
I
....--- T
- ' ~

5 Write in English language the following symbolic propositions.

1
· 'i) (3x)(Vy) R (x, y) , (ii) ~ l(Vx) Q (x)] (M u
, . . 2004)
l Ans. : (i) (3x) (\iy) R (x, y) means for all y there exists an x such that x and Y are related.
~
(ii) N [(\/x)Q(x)l means (:lx) O(x) means there exists an x such that Q(x) Is false. ]
16_Write English sentences corresponding to the following .
(i) (v'x)(3y)R(x, y) , (ii) (3x)(v'y)R(x,y ), (iii) (Vx)[rvQ (x)l,
(iv) (3y)l~ P (y)L (v) (v' x) P (x)

where P(x) ·· xis even; 0 (x) : xis a prime, R (x, y) : x + y is even. (M.U. 2005)
l Ans.: (i) (v'x)(3y) R (x, y) means for all x there exists a y such that x + y is even .
(ii) (3x)(\iy) R (x, y) means for all y there exists an x such that x + y is even.
(iii) (\1' x) l~ Q (x)l means for all x, x not a prime.
(iv) (3y)l~ P (y)1 means there exists a y such that y is not even.
(v) (\1' x) P (x) means for all x, xis even. 1
================~{1.%\~i-S!§~:::e:::x!JJ\:Js:-r-~=-==-=====--=====-======::::===-====-====-~:::::~
Use Mathematical Induction to prove.
(A) 1 .. 1 + 5 + 9 + .. ~ .. + (4n - 3) = n (2n - 1)
(M.U. 2012)
2. 2 + 4 + 6 ..... + 2n = n (n + 1)
n(n+1) n
3. 1+ 3+6+ ..... + - - - = --(n+1)(n+2)
2 6
. n{:;n + 1)
4. 2 1 5 + 8 + ..... + (3n -1) =- -- (M.U. 199n
2
2 2 2 n
5. 1 +3 +5 + ..... +(2n-1) 2 =-(2n-1)(2n+1)=--- (4n3 - n)
(M.U. 1999, 2013, 13)
3 3
3 3 3
6. 1 + 3 + 5 + ..... + (2n - 1)3 :;: n 2 (2n 3 - 1)
7. i 1+ 22 + 23 + ..... + 2n == 2n+ 1 - 2
2
8. 1 + 2 2 + 3 2 + ..... + n2 = n (n + 1)(2n + 1)
6 -
9. _1 + _1 _1 . 1 1
21 i2- + 23 + ..... + 2" =1 - 2n
1 o. 1 • 2 + 2 • 3 + 3. 4 + ( n
· .... + n • n + 1) === - (n + 1)(n + 2)
1 1 1 3
11.--+--+--+ 1 n
3•7 7•11 11•15 ..... + - : - - : - - - - - - = - - -
(4n -1)• (4n + 3) 3(4n + 2)
Jlscrete iv1au i t: 11,au'- »

1 1 1 1 _ n
12· 2-5 + 5 • 8 + 8 • 11 + · · · · · + (3n - 1) • (3n + 2) - 2 (3n + 2)

13. (1- i )" = 2" ( cos n 1t - i sin n 1t)

12
4 4

14. a+(a +d)+ (a+2 d)+ ...... +[a+ (n-1) d]= ;[2a+ (n-1) d]
2
B) 1. Prove that 4 n - 1 is divisib le by 15.
3
2. Prove that 2 n - 1 is divisib le by 7.
3. Prove that gn - 2n is divisib le by 7.
4. Assum ing log mn = log m + log n, prove that log x n = n log x.
( Hint : P (k + 1) = log x'< + = log (x'< · x)
1

+
= log JI + log x = k log x log x = ( k + 1) log x )
2
6. Prove that 2n+ + a2 n+ 1 is divisib le by 7.
7. Prove that 3n+
2
- Bn - 9 is divisib le by 64. (M.U. 2006)

8. Prove that 2n + 9n + 13n + 7 > 0 for all n ~ - 2.

3 2

9. Prove that 2n < n I for all n ~ 4.

1 O. Prove that 2 · 7n + 3 · 5n - 5 is divisib le by 24 for all n> 0.

11. Prove that 3n - 2n - 1 is divisib le by 4 where n E N. (M.U. 1996, 97)

(M.U. 1997)
12. Prove that 5n + 3 is divisib le by 4.
(M.U. 1998)
13. Prove that one of the integer s a, a+ 2, a+ 4 is divisib le by 3 where a E
/.

coso sinOln [ cos no sin nOl

14. Prove that
[ -sinO cos e J -- sin n e cos noj

15. Prove that LO J _Li/01 nn]

11 11/n
16. If y = etan-' x then prove that (1 + x2) Yn+1 + (2nx-1 ) Yn + (n -1) Yn-1 == 0.

17. lfy=xl og(x+ 1)then prove that Yn =(-1t -2. (n-2) !(x+n )
(x + 1 )n