Jaypee Institute of Information Technology (JIIT) 2007 B.Tech MATH --DISCRETE MATH - Question Paper

Tutorial Sheet – IV (Logic, Propositional functions & quantifiers)
1. (a) Which of the subsequent phrases are propositions:
(i) Is this true?
(ii)
(iii) ans this ques..
(iv) five + six = 12
(v) 4 is even.
(b) What is the negation of: (c) Determine the truth value of the following:
(i) Today is Tuesday. (i) six + two =7 or four + four = 8
(ii) five + one = 6. (ii) three +1 = four and five + four = 7
(iii) No 1 wants to buy my house. (iii) four + three = seven and six + two = 8
(iv) a few students have no mobile phone. (iv) If three * five = 24 then three + five = 8
(v) Every even integer greater than four is the sum (v) If three * five = 15 then three + five =12
of 2 primes.
2. Consider the following:
p: This computer is good.
q : This computer is cheap.
Write every of the subsequent statements in symbolic form:
(i) This computer is good and cheap.
(ii) This computer is not good but cheap.
(iii) This computer is costly but good.
(iv) This computer is neither good nor cheap.
(v) This computer is good or cheap.
3. Consider the following:
p : you take a course in Discrete Mathematics
q : you understand logic.
r : you get an A+ in this course.
Write in simple phrases the meaning of the following:

4. Construct the truth table for the following:

5. Determine whether the subsequent propositions are tautologies or not.

6. Show that the propositions are contradiction.
7. Show that the subsequent pairs of propositions are logically equivalent:

8. State the converse, inverse and contrapositive of the following:
(i) If today is Easter then tomorrow is Monday
(ii) If John is a poet then he is poor.
(iii) If triangle ABC is right angled then
(iv) If P is a square then P is a rectangle.
(v) If a triangle is not isosceles then it is not equilateral.
(vii) If the square of an odd integer is odd then that number is odd.
9. Write the negation of every statement as simply as possible.
(i) If she works, she will earn money.
(ii) He swims if and only if the water is warm
(iii) If it snows, then they do not drive the car.
10. Determine the validity of the subsequent arguments.

(iv) If I study then I will pass in exam.
If I don’t go to cinema, then I will study.
But I failed in exam.
…………………………….
Therefore I went to cinema.

11. Let A = {1, 2, 3, 4, 5}. Determine the truth value of every of the subsequent statements:

12. Determine the truth value of every of the subsequent statements where U ={1, 2, 3} is the universal set:

13. Negate every of the subsequent statements:

14. Let p(x) denote the sentence “x+2 > 5”. State whether or not p(x) is a propositional function on every of the subsequent sets:
(a) N, the set of positive integers (b) M = {-1, -2, -3, …} (c) C, the set of complex numbers
15. Negate every of the subsequent statements:
(a) All the students live in the hostels.
(b) All mathematics majors are male.
(c) Some students are 18 (years) or older.
16. Let A = {1, 2, 3, …,9, 10}. Consider every of the subsequent phrases. If it is a statement, then determine its truth value. If it is a propositional function, determine its truth set.

17. Negate every of the subsequent statements:
(a) If the teacher is absent, then a few students do not complete their homework.
(b) All the students completed their homework and the teacher is current.
(c) Some of the students did not complete their homework or the teacher is absent.
18. Find a counterexample for every statement where U = {3, 5, 7, 9} is the universal set:

19. Negate the statement .

Earning:   Approval pending.