B3B036S Pages: 2

PART D
Answer any 2 Questions, 9 marks each.
12. For the set I= {0,1,2,3},show that modulo 4 system is a ring. (9)
13. a) Prove that the order of each sub group of a finite group G is a divisor of the order of group
G. (6)
b) IfA = (1234 5) and B= (2 3)(4 5). Find product of permutation AB. (3)
14. If (L, =) is a lattice, then for any a,b,c eL, the following properties hold. If b > ௦,
11001) ೩೫ 58೩1೪೫೦ ii)a*% bPARTE

Answer any 4 Questions, 10 marks each.

15. (a) Show that (P—Q) a (QP) is logically equivalent to PQ. (5)
(b) Suppose x is a real number. Consider the statement "If x’ = 4, thenx = 2."
Construct the converse, inverse, and contrapositive (5)
16. (a) Prove that (PAQ) — (PQ) is a tautology. (5)
(b) Show that (a v b ) follows logically from the premises
ഡു, 1 (1 - ‏رج جرد‎ -> (saat) and (sant) -> (३.४४ ) (5)
17. (a) Represent the following sentence in predicate logic using quantifiers 1) All men are
mortal. ii) Every apple is red iii) Any integer is either positive or negative. (6)
(b) Use the truth table to determine whether p —(qalq) and Ip are logically equivalent.
(4)

18. Show that the premises “one student in this class knows how to write programs in JAVA”
and “Everyone who knows how to write programs in JAVA can get a high paying job” imply
the conclusion “Someone in this class can get a high paying job” 10

19. (a) Prove the following statement by contraposition:

If a product of two positive real numbers is greater than 100, then at least one of the
numbers is greater than 10. (6)
(b) Express the negation of the following statement in English using quantifiers
i) If the teacher is absent, then some students do not keep quiet ii) All students

keep quiet and teacher is present. (4)

20. (a) Prove that V2 is irrational using proof by contradiction. (6)

(b) Find the truth table (~ Q >~P) > (P > Q) (4)
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