PREPARING FOR THE

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PREPARING FOR THE
PART II - Mathematics for Computing I
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Mathematics for Computing I

• Today we will discuss some
• Frequently Asked Questions
• Model Paper Questions on Logic
• namely 26, 28, 30, 32, 38

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Answers to Frequently Asked Questions about
Mathematics for Computing I
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• How to clarify any doubts?
• E-mail us, the address is
• mc_at_ict.cmb.ac.lk

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When can we have the Model Paper?

• Model Papers are available in the web
• www.ict.cmb.ac.lk/bit.htm
• Also they have been posted to you.

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• Model Question 26
• Consider the following
• 2 is not an integer
• Is 2 a positive integer?
• The presidential system in Sri Lanka was
  abolished in the year 2000
• x2 gt 10

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• Which of the following are (is) correct?
• (a) (i), (ii), (iii) are propositions
• (b)  (i), (iii), (iv) are propositions
• (c)  (i), (iii) are propositions
• (d) None of (i) (ii), (iii), (iv) are
  propositions
• (e)  All (i), (ii), (iii) (iv) are propositions.

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• Objectives
• To recognise sentences which are propositions and
  those which are not propositions.

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• Solution
• Definition
• Something written is called a proposition if it
  is either true or else it is false.

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Solution...

• 2 is not an integer
• Is a proposition. It is false.
• Is 2 a positive integer?
• Is not a proposition as it is a question and we
  cannot talk of it being true or it being false.
• The presidential system in Sri Lanka was
  abolished in the year 2000
• Is a proposition. It is false.

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Solution...

• x2 gt 10
• Is not a proposition. In x2 gt 10 we do not know
  the value of x. So we cannot say whether x2 gt 10
  is true or whether it is false.
• E.g. if x takes the value 2 it is false, but if x
  takes the value 4 it is true.
• So, we have (i), (iii) are propositions and
• ( ii), (iv) are not propositions.
• So, (c) is the only correct choice.

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Which is tested in Mathematics - the Theory or
its application ?

• Both.
• The students have to know the theory as well as
  their applications.
• Refer module objectives or URL
• www.ict.cmb.ac.lk/bit.htm

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Are we allowed to use calculators for the
Mathematics Paper ?

• No.
• We are not testing the calculating knowledge of
  the students. You need to study the theory and
  its various applications.

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• Model Question 30
• Out of the 8 possible truth values for p, q, r
• (p?q) ? r
• is true only for
• (a) 7 set values (b) 6 set values
• (c) 5 set values (d) 4 set values
• (e) 3 set values

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• Objectives
• To know the truth tables in summary and thus to
  apply them fairly quickly.

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• Solution
• (p?q) ? r is F only when p?q is T with r is F
• i.e. it is F only in the following cases
• p is T, q is T, r is F
• p is T, q is F, r is F
• p is F, q is T, r is F
• i.e. it is F only for 3 sets of values
• ? It is T only for 5 sets of values

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Announcement

• BIT ID cards and admission cards are being posted.

• If you do not receive them by 20/03/2001, please
  contact the EEU of ICT.
• Tel. [phone]

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• Model Question 28
• Consider the following.
• (i) p?p (ii) p?((p?q)) (iii) p?(p?q)
• Which of the following is correct?
• (a) (i), (ii), (iii) are all tautologies.
• (b)  (i), (ii) are tautologies but (iii) is not a
  tautology.
• (c)  (i), (iii) are tautologies but (ii) is not a
  tautology.
• (d)  (i) is a tautology but (ii), (iii) are not
  tautologies.
• (e)  None of (i) (ii), (iii), (iv) is a tautology.

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• Objectives
• To know the truth tables in summary so that
  truth values of compound propositions are found
  out very quickly.

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• Solution
• Definition
• A proposition which in all circumstances is
  true, is called a tautology.

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Solution...

• It is immediately seen that
• i.e. (i) (p?p ) is a tautology, since one of p,
  p is T. (In a truth table for ? if at least
  one of the components is T then the ?
  proposition is T).
• p ? (p?q) is F when p is T and q is F. (Since we
  get p T, p?q F).
• So (iii) is not a tautology.

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Solution...

• When p is T, p ? ((p ? q)) is T.
• When p is F, p ? q is F and so (p ? q) is
  T.
• ? when p is F, p ? ((p ? q)) is T
• ? In all circumstances p ? ((p ? q)) is T
• ? (ii) is a tautology.
• So, the only correct choice is (b)

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Solution...

• Although, what is written on the previous slides
  is long, the thinking behind it is short and
  fast.
• There is another long but sure way of answering
  this question. This is to draw the truth tables
  of (i), (ii), (iii).

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Solution...

• It is seen from this truth tables that only (i)
  and (ii) are tautologies. So the only correct
  choice is (b).

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• Model Question 32
• The following four propositions are got by
  substitutions p, q, r, p, q in (i), (ii),
  (iii), (iv) for the propositions there which are
  connected by and, or, but
• (?) p?r (?) r?q
• (?) (q)?p (?) r?p

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• Model Question 32...
• Which of the following is correct?
• (i) It rained yesterday and there is no play at
  the Royal_ Thomain match today,
• (ii) The ground is wet or it rained yesterday.
• (iii) It rained yesterday and the ground is dry.
• (iv) There is play at the Royal_Thomian match
  today but the ground is wet.

(?) p?r ( ? ) r?q ( ? ) (q ) ?p ( ? )
r?p
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• Model Question 32...
• Choices
• (a) (i) is (?) , (ii) is (?), (iii) is (? ) and
  (iv) is (?)
• (b) (i) is (? ) , (ii) is (?), (iii) is ( ?) and
  (iv) is (?)
• (c) (i) is (?) , (ii) is (?), (iii) is (? ) and
  (iv) is (? )
• (d) (i) is (?) , (ii) is (?), (iii) is (? ) and
  (iv) is (? )
• (e) (i) is (? ) , (ii) is (?), (iii) is (? ) and
  (iv) is (?)

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• Objectives
• To recognise the logical form of sentences given
  in ordinary English
• To learn to think logically in solving problems.

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Solution
(?) p?r (ii) The ground is wet or it rained
yesterday.

• (ii) is (?)
• (This is seen by the fact that only sentence with
  an or is (ii). This is also seen by the fact
  that in all the choices we have, (ii) is (?)
• r appears in (?). ? r appears in (ii)
• ? r is one of the ground is wet, it rained
  yesterday

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Solution...
(?) r?q (?) r?p
(i) It rained yesterday and there is no
... (ii) The ground is wet or it rained
yesterday. (iii) It rained yesterday and the
ground is dry. (iv) There is ... but the ground
is wet.

• r appears again twice and in both places there is
  and.
• ? r must be, it rained yesterday
• Since (?) is p?r, p must be the ground is wet
• Since we know p, r we get that (iii) is (? )

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Solution...
(?) (q)?p (iv) There is ... but the ground is
wet.

• the ground is wet appears in (iv). ?(iv) is
  (?)
• (Note but means here the same as and. In
  English but means a bit more than and.
  Example He is young but his hair is gray. We
  can say He is young and his hair is gray but
  earlier sentence is saying more than this)
• ? the correct choice is (c)

(c) (i) is (?) , (ii) is (?), (iii) is (? ) and
(iv) is (? )
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• Model Question 38
• It is given that p ? q and p ? q are true. Now
  which of the following must necessarily be true,
• (a) p (b) p (c) q
• (d) q (e) q ? r

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• Objectives
• Learn logical arguments.
• i.e. given premises (i.e. propositions taken to
  be true) to derive valid conclusions

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• Solution
• One of q, q is F. Since p ? q, p ? q are taken
  to be true, p must be F
• i.e. p must be true
• with p is F, p ? q, p ? q is T both when q is T
  and when q is F (i.e. q is T)

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• Solution
• when q is F, q ? r is T
• when q is T and r is F, q ?r is F
• ? only, p is necessarily true
• ?(b) is the only correct choice

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• Solution
• we can also do this by drawing truth tables in
  the following manner

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• Solution
• From the truth table we get that P must be F
• i.e. p must be T
• q could either be T or it could be F
• So, we cannot say that q must be T

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• Solution
• Also we cannot say, q must be T
• we can have q is T, r is T. When this happens q ?
  r is T
• We can also have, q is T, r is F. When this
  happens q ? r is F

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• Solution
• ? only p is necessarily true
• ? (b) is the only correct choice
• Note From what we have written in the first
  method, it appears long. However, this can be
  done mentally and then it is quite short.
• Also, importantly, that method involves logical
  thinking