Discrete Mathematics Answers

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.

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.

Which of the following are propositions? Of those which aren’t, explain why.

The baby is laughing.

What’s that noise?

1 + 1 = 3

Get out of here!

He ran in the race but slipped on a banana peel.

What a glorious day it is!

Santa Claus is jolly.

To fail to achieve the impossible is not to fail.

Please pay attention.

All triangles have four sides.

The next Prime Minister will be a woman.

What are you thinking?

If it rains then there is moisture in the air.

I wish to be immortal.

Neither circumstances nor criticism will prevent my progress.

Won’t you close the door?

Santa Claus is a fictional character.

Let x mark the spot.

Your wish is my command.

Define ‘proposition’ to mean ‘bearer of one truth’.