Answer to Question #250308 in Discrete Mathematics for Toby

Question

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.

Answer 1

1. It is not hot.

$\lnot$ p

2. It’s hot and it’s wet.

p$\land$ q

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

$\lnot$ q$\land\lnot$ r

4. It’s not both hot and wet.

$\lnot$ (p$\land$ q)

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

r$\land$ p$\land$ q

6 I am not tired but it is wet.

$\lnot$ r$\land$ q

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

( $\lnot$ p$\land\lnot$ q)$\land$ r

8. It’s not the case I am not tired.

$\lnot$ r

9. Either I am tired or it’s hot.

r$\lor$p

10. I am tired or it’s hot.

r$\lor$p

11. It is wet or it’s not hot.

q$\lor\lnot$ p

12. It’s either not wet or not hot.

$\lnot$ q$\lor\lnot$ p

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

$\lnot$( p$\lor$q)

14. It’s hot or wet but not both.

( p $\lor$ q)$\lnot$ (p$\land$ q)

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

p$\lor$q$\lor\lnot$ r

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

r$\land\lnot$ p$\lor$ q

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

( p$\land$ q)$\lor$ ( p$\land\lnot$ q)

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

$\lnot$ q$\lor$ (p$\land$ q)

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

$\lnot$ r $\land\lnot$ p

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

$\lnot$ (p$\land$ q)$\lor$ (p$\land$q)