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B B3B037 Total Pages:3

PART E
Answer any four Questions. Each Question carries 10 marks
15. a. Without using truth tables, prove the following (ನ. P VQ) A (PA (PA Q)) =P AQ
b. Show that ((P -> Q) A(Q ೫)) -> (? -> R) is a tautology.

16. a. Convert the given formula to an equivalent form which contains the connectives
sand Aonly: 7(P#(Q (೫ V P)))
b. Show that = PA (“QA ۶(۷ (QAR) V (PAR) => २.
17. a. Show that 5 V R is tautologically implied by (P ५ Q)A(P > R ) A(Q—S)

b. Prove the validity of the following argument “If 1 get the job and work hard, then I
will get promoted. If 1 get promoted, then I will be happy. I will not be happy.
Therefore, either I will not get the job or I will not work hard.

18. a. Prove that (Ax)(P(x) AQ(x)) >(4x)(P(X)A(Ax (Q(x)

b. Consider the statement “Given any positive integer, there 15 a greater positive

integer’. Symbolize this statement with and without using the set of positive integers

as the universe of discourse.
19. a. Show that R A (0 ५ Q) is a valid conclusion from the premises PVQ,Q—-R,
? -+ ۷۸۲ 0 - ५.

0. Prove by mathematical induction that 7൦ is divisible by 43 for each

positive integer n.

20. Discuss indirect method of Proof. Show that the following premises are inconsistent.
(i) If Jack misses many classes through illness, then he fails high school.
(ii) If Jack fails high School, then he is uneducated.
(iii) If Jack reads a lot of books, then he is not uneducated.

(iv) Jack misses many classes through illness and reads a lot of books.

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