# Set problems(Of which one includes propositional logic)

1. Oct 17, 2012

### Meneldur

I have a few questions regarding 2 set problems.

Exercise 1:
1. The problem statement, all variables and given/known data
1. the set A = P(empty) (the powerset of the empty set);
2. the set B = P(A);
3. the set C = P(B).

2. The attempt at a solution

1. A= {empty}
2. B = {empty, {empty}}
3. C = {empty, {empty}, {{empty}}, {empty, {empty}}}

Exercise 2:
1. The problem statement, all variables and given/known data
Formalise the Dolphin puzzle in set theory.
Use chaining of set inclusions to derive the conclusion.
Puzzle:
(1) The only animals in this house are cats.
(2) Every animal is suitable for a pet, that loves to gaze at the moon.
(3) When I detest an animal, I avoid it.
(4) No animals are carnivorous, unless they prowl at night.
(5) No cat fails to kill mice.
(6) No animal ever take to me, except what are in this house.
(7) Kangaroos are not suitable for pets.
(8) None but carnivora kill mice.
(9) I detest animals that do not take to me.
(10) Animals that prowl at night always love to gaze at the moon.
- Argue that they imply I always avoid a dolphin.

2. The attempt at a solution
Sadly I don't even know how to attempt to solve this so any help is greatly appreciated :)

Cheers.

2. Oct 18, 2012

### Meneldur

After some further study I've come to a few ideas about how to solve it:
Animal ⊆ Cat
Gaze ⊆ Suitable
Detesh ⊆ Avoid
Carnivorous ⊆ Prowl
Cat ⊆ NotFail
Take to me ⊆ Animal
Kangaroo ⊆ NotSuitable
Carnivora ⊆ KillMice
NotTake ⊆ Detest
Prowl ⊆ Gaze
Now i'm not even sure of those are correct. But assuming they are how do I draw a conclusion out of them? And do I need to define them as sets first?

3. Oct 18, 2012

### HallsofIvy

Staff Emeritus
Since the objective is to arrive at "I always avoid a dolphin" I would start by looking at "(3) when I detest an animal I avoid it" (the only one that involves "I detest") as the last line so the problem becomes showing "I detest dolphins". However, I see NO statement that even mentions "dolphins".
If the statement said "kangaroos" rather than "dolphins", then it would be straight forward.

4. Oct 18, 2012

### Meneldur

Oh sorry about that, I don't know what i was thinking about while I was typing that. Kangaroo = dolphin