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Build a complete truth table and Show that they are Logically equivalent

¬(¬P ∧ Q) ∧ (P ∨Q)≡ p


Consider the following argument: I will get grade A in this course or I will not graduate If I do not graduate, I will join the army I got grade A. Therefore, I will not join the army Is this a valid argument?
Prove that for any propositions p,q,r the compound proposition {pà ƒ ƒ ¢ † ’(qà ƒ ƒ ¢ † ’r)}à ƒ ƒ ¢ † ’{(pà ƒ ƒ ¢ † ’q)à ƒ ƒ ¢ † ’(pà ƒ ƒ ¢ † ’r)} is a tautology
Find the characteristic root of the recurrence relation an=anà ƒ ƒ ¢ ˆ ’1+2anà ƒ ƒ ¢ ˆ ’2.
Let {an} be a sequence that satisfies the recurrence relation an=anà ƒ ƒ ¢ ˆ ’1+3anà ƒ ƒ ¢ ˆ ’2, for n=2,3,.,.,., where a0=1,a1=2. Find the values of a2,a3.

Solve the following:

(a) 1231001 (mod 101)

(b) 17123 (mod 13)



Let a and b be two Natural Numbers, such that the greatest common divisor of a and b is 63, and the least common multiple of a and b is 44452800. If ’b’ is an odd number, what is the minimum value of ’a’ possible? [Hint: a · b = gcd(a, b) · lcm(a, b)]



1. Convert each of the following to their respective Decimal, Octal, Hexadecimal and binary representation:

(a) (742)8

(b) (1011)2

(c) (47)10

(d) (3EAC)16

If 2 ≥ 3, then the cube of -1 is -1



Translate the following into symbols, given;

p: It is hot.

q: It is wet.

r: I am tired.

Example: 13. It is not the case that it is either hot or wet.  ~(p∨q)

1. It is not hot.

2. It’s hot and it’s wet.

3. It is not wet and I am not tired.

4. It’s not both hot and wet.

5. I am tried and it is hot and it’s wet.

6 I am not tired but it is wet.

7. It’s neither hot nor wet, but I am tired.

8. It’s not the case I am not tired.

9. Either I am tired or it’s hot.

10. I am tired or it’s hot.

11. It is wet or it’s not hot.

12. It’s either not wet or not hot.

13. It’s not the case that it’s either hot or wet.

14. It’s hot or wet but not both.

15. It’s hot or wet or I am not tired.

16. I am tired but it’s not hot or wet.

17. It’s hot and wet or it’s hot and not wet.

18. Either it’s not wet or it’s both hot and wet.

19. It’s not the case that, I am tired but it’s not hot.

20. Either it’s not both hot and wet, or it is.
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