PUCS3BS06-MATHEMATICS FOR COMPUTER SCIENCE AND ENGINEERING
UNIT-2 : MATHEMATICAL LOGIC
2 Marks
1. Define Propositions and give example.
2. Define Conditional Statement and write Truth Table.
3. Give the converse, inverse and contra positive for the following “If Sita is rich, then she is happy”. [AU N/D'15]
4. What is the contra positive of the statement “the home team wins whenever it is raining”? [AUN/D'17]
5. Write the negation of the following statement “It is not raining and I will not take the umbrella”. [AU N/D'11]
6. Prove that P → (P V Q) is a tautology. [AU N/D'22]
7. Prove that 7p → (q → r) and q → (p V r) are logically equivalent. [AUN/D'17, N/D'10]
8. Show that P → (Q V R) ↔ (P → Q) V (P →R) using truth table. [AUM/J'17]
9. Find PCNF of (P ˄ Q) V (7P ˄ Q). [AU N/D'12]
10. What is the negation of the statement (i) (Ɐ x) ((𝑥 > 𝑥) and (ii) (∃x) (𝑥 = 2)? [AU N/D’13]
11. Negate the following proposition 1. (∃ x) (P(x)˄Q(x) 2. (x)(P(x) V Q(x)).
12. Symbolic the following expressions (1). “Everyone has mother”(2). “Some x married to some y”.
13. Express the negation of the following statement using quantifiers and in English:
“No one has done every problem in the exercise”
14. Symbolic the expression, “all the world loves a lover”.
15. Prove that 7(∃y) (x) P (x, y) ↔ (y)( ∃x) 7P (x, y).
16. Negate the following proposition (∃x) (Ɐ y) p (x, y).
17. Negate the following proposition (x) (∃x) (P (x, y) → Q (x, y))
18. Write the converse, inverse, contra positive of ‘If you work hard the you will be rewarded’.
19. When do you say those two compound statement propositions are equivalent?
20. Obtain the disjunction normal forms of p ˄ (p→ q).
8 Marks:
1. Prove the following implication by using truth table.(P → Q) ˄ (Q → R) => (P →R) using truth table. [AU N/D'15]
2. Show that the following equivalence are true without using truth table. P→ (Q → P) ↔ 7 P → (P → Q) [AU A/M'11]
3. Show that (7P ˄ (7Q ˄ R)) V (Q ˄ R) V (P ˄ R) ↔ R, without using truth table. [AUA/M'23]
4. Show that (P V Q) ˄ 7 (7P ˄ (7Q V 7R)) V (7P ˄ 7Q) V (7P ˄ 7R) is a tautology using truth table. [AU A/M'18]
5. Obtain the principle disjunction normal form for P → [(P → Q) ˄ 7(7Q V 7P)]. [AU N/D'22]
6. Without using truth table, find the PDNF and PCNF of [P → (Q ˄ R)] ˄ (7P → (7Q ˄ 7R)). [AU A/M'11]
7. Show that R ˄ (P V Q) is a valid conclusion from the premises P V Q, Q → R, P → M, 7M. [AU N/D'16, A/M'11]
8. Show that hypotheses “If is not sunny this afternoon and it is colder than yesterday”.“If we will go swimming, then it is sunny”.
“If we do not go swimming, then we will take a canoe trip” and “If we take a canoe trip, then we will be home by sunset” lead to
the conclusion “We will be home by sunset”.
9. Prove that A → 7D is a conclusion from the premises A → (B V C), B→ 7A and D → 7C by using conditional proof. [AU
A/M'11], N/D'11]
10. Prove that the argument is valid p → 7q, r → q, r => 7p [AU M/J'12]
11. Show that (∃x) [F(x) ˄ S(x)] → (y) [M(y) → W) (y)], (∃y) [M(y) ˄ 7W(y)] => (x)[F(x) → 7S(x)] [AU N/D'10]
12. Establish the validity. “All integers are rational numbers”, “Some integers are power of 2”, Conclusion: “Some rational
numbers are power of 2”. [AU N/D'22]
13. Show that (Ɐx) [P(x) V Q(x)] => (Ɐx) P(x) V (∃x) Q(x). [AU A/M'23, '18]
14. Verify that validity of the following inference. “If one person is more successful than another, then he has worked harder to
deserve success”. “Ram has not worked harder than Siva”. Therefore, “Ram is not more successful than Siva”. [AU A/M'11]
15. Verify that validity of the following arguments. “Every living thing is a plant or an animal”. “John’s gold fish is alive and it is
not a plant”. “All animals have hearts”.Therefore “John’s gold fish has a heart”. [AU M/J'12]
