UNIVERSITY OF COLOMBO

1
UNIVERSITY OF COLOMBO SCHOOL OF
COMPUTING
IT1101 Mathematics for Computing-I
DEGREE OF BACHELOR OF INFORMATION TECHNOLOGY
2
9
Consider the following truth tables.
i)
ii)
iii)
Which one of the following is correct?
3

• (a) (i), (ii) and (iii) are all correct.
• (i), (ii) are correct but (iii) is incorrect.
• (i), (iii) are correct but (ii) is incorrect.
• (ii), (iii) are correct but (i) is incorrect.
• (e) (i), (ii) and (iii) are all incorrect.

We will first look at each of (I), (ii), (iii)
to see wheather it is correct or not. Then we
will be able to decide which one of (a), (b),
(c), (d), (e) we have to choose
4
A proposition of the form F gt? is F only when F
is T while ? is F ---- (a) So, P gt(P gtQ) is F
only when P is T while PgtQ is F. Now Pgt Q is F
only when P is T while Q is F. ?Pgt (PgtQ) is F
only when P is T and Q is F ? (i) is correct
5
P v not Q is F only when both p is F and not Q
is F ?P v not Q is F only when both P is F and Q
is T ? (ii) is correct
6

• By (a), Pgt(P?Q) is F only when both P is T and P
  ?Q is F But when P is T, P ? Q is F only when Q
  is F
• ?P gt (P ?Q) is F only when P is T and Q is F
• (iii) is correct
• So our choice is (a)

7
12)
Consider the following (i)                (P
? (not P)) ? (P ? (not P)). (ii)              (P
? (not Q)) ? (Q ? (not P)). (iii)             P?
(Q? (not P)) ? ((not Q) ? (not P)).   Which of
the following is/are correct?
8
(a)    (i) is a contradiction. (b)    (iii) is a
tautology. (c)    (ii) is neither a contradiction
nor a tautology. (d)    (iii) is neither a
contradiction nor a tautology. (e)    (ii) is a
contradiction.
9
Consider a proposition made from some
propositional variables. Then, it is a tautology
if it is true for every combination of truth
values of the propositional variables. It is a
contradiction if it is false for every
combination of truth values of the propositional
variables.
10

• (P ? not P) gt (P ? not P) is made from only one
  propositional variable, and it is P. We know that
  P ? not P is always true (ie it is a tautology)
  and that P ? not P is always false (ie it is a
  contradiction)
• (P ? not P) gt ( P ? not P) is always F (see (a)
  of the answer to question 9)
• It is a contradiction

11

• (a) is correct
• If you did not know what was mentioned above you
  would have to consider the case, P is T, and the
  other case, P is F. In both cases you will find
  that P ? not P is T while P ? not P is F . So you
  would conclude that
• (P ? not P) gt (P ? not P) is always F and here
  that it is a contradiction

12

• Let us look at (ii) It is of the form ? ??.
• ?? is F only when ? ,? have different truth
  values ------ (?)
• Also, not (? ? ? ) has the same truth value as
  (not ?) ? (not ?) ----(?)
• ? Not (P ?(not Q) has the same truth value as not
  P ? not (not Q)

13

• Not (P ?(not Q) ) has the same truth
• value as Q ? not P
• P ? not Q, Q? not P have have different truth
  values
• By (?) (P ? not Q) ? (Q? not P ) is always F and
  hence it is a contradiction
• (c) is incorrect and (e) is correct

14
Even if you are not aware of (?) you could
answer this in the following way Consider the
case, P ? not Q is T , and the other case, P ?
not Q is F. Then see that in each case P ? not Q,
Q? not P have opposite values and so by (?) ,
you get that it is a contradiction
15

• Now consider(iii)
• Since it is of the form ?1 ? ?2 ? ? 3 We see at
  once that when P is T,
• P ?(Q ? not P) ? ((not Q) ? not P) Is T.
• When P is F, not P is T. Also one of Q, not Q is
  T.
• when P is F, one of Q ? not P,(not Q) ? not P is
  T.
• When P is F,
• P ? (Q ? not P) ? ((not Q) ? not P)is T.
• It is a tautology.

16
?(b) is correct and (d) is incorrect. So we have
our answer.
16)
For n ?N, P(n) is a proposition such that for any
n ? N, P(n 1) ? P(n) is true.  Which of the
following are(is) true?
17
(a) For any n ? N, P(n) (b) For any n ? N, P(1)
? P(n) (c) For any n ? N, (P(5) ? n ? 5) ?
P(n) (d) P(10) ? P(9) (e) P(9) ? P(10)
WE note that instead of having, for any n ? N,
P(n) gt P(n1), what we have here is, for any
n ? N, P(n1)gtP(n)
---(?)
18
Consider (a)(a) for any n ?N ,P(n) We see that
if P(n) is F for every n ? N, then (?) will be T.
So we cannot say that for any n ?N, P(n), is
T So (a) is incorrect.
19
Consider (b) (b) for any n ?N P(1)
gtP(n) If P(1) is F, P(1) gtP(n) is always
T. But we can have P(1) is T, while for every n
?N\1, P(n) is F. So that for every n ?
N\1, P(1) gtP(n) is F. When this happens (?) is
T. So (b) is incorrect.
20
Consider (c) when P(5) ? nlt5 is F, (P(5) ?
nlt5) gtP(n) is T. Suppose P(5) ? nlt5 is T. Since
P(5) is T by (?) (i.e. for any n ? N ,
P(n1)gtP(n), We get P(4) , P(3), P(2), P(1) are
all T. i.e. when nlt5, P(n) is T.
21
So, when P(5) ? nlt5 is T, P(n) is T and hence
(P(5) ? nlt5)gtP(n) is T. ? (c) is correct.(i.e.
(c) is true) Consider(d) P(10) gt P(9) follows
immediately from (?)i.e. for any n ?N,
P(n1)gtP(n) So (d) is correct.
22
Consider (e) when P(1), P(2) P(9) are all T
while P(n) is F whenever n?10 , (?) i.e. for any
n ?N, P(n1) gtP(n) is T. But when this happens,
P(9) gtP(10) is F So (e) is incorrect.
23
1)
Which of the following sets is(are) null (i.e.
empty)? (a) x x ? Z and x ? 0 (b) x x ?
N and 15 ? x ? 16 (c) x x ? Z and -16 ? x ?
-15 (d) x x ? Z and x ? -3 (e) x x ?
N and x is prime and x is even
24
N1,2,3,14,15,16,17, Z..
17,-16,-15,-14,.-2,-1,0,1,2,3,. It is seen
that the sets in (a) and (d) are non-empty.
Between 15 and 16 there is no positive integer,
and so the set in (b) is empty. Between 16 and
15 (-15-161) there is no integer. So the set
in (c) is empty This leaves only the set in (e).
Now 2 is prime and even. So the set in (e) is
non-empty so the correct choices are (b),(c)
25
3)
Consider the following Venn diagram, where the
sets A, B are represented by the regions bounded
by the egg-shaped closed curves and the regions
1,2,3,4 are as indicated.
B
A
1
2
3
4
26
In the above Venn diagram, Ac ? Bc is represented
by (a) the region 2 (b) the region 3 (c) the
region 1, 4
(d) the region 1, 3, 4 (e) the region 4
We first note that Ac ? Bc corresponds to one
definite region. So , only one choice is correct
27
(A ? B)c Ac ? Bc (?) ( from DeMorgans
laws)The region for A ? B is 2 and by (?), the
region for Ac ? Bc is 1,3,4 So the correct
choice is (d) Another way of doing this is A
1,2 (i.e. the region for A is 1,2) B 2,3. Ac
3,4, Bc 1,4 ?Ac ?Bc 1,3,4.
28
Thank you