**Tasks:**

1) Translate (i) - (vii) into English

2) Figure out who is in the cabal

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

**Solution:**

Task 1)

(i)

There are several options how to translate this into English:

i.a) There exist x, y, and z. None of them are the same towards each other. And each of them is in cabal.

i.b) There exits (at least) 3 different people who are in cabal.

The two English sentences are equivalent towards each other. The reason I put "at least" in brackets is because adding "at least 3" adds more information to it. If you have written "exactly 3", then you are wrong. Because the expression mentioned 3 elements to be valid, i.e. x, y, and z. If there is a fourth, fith, sixth, etc. element that is also valid, then the expression still holds for these additional elements. Therefore "

**at least 3**" is the correct translation.

If you wanted to write

**exactly 3 elements**, then you could have written something like this:

If you wanted to write

**at most 3 elements**, then you could have written something like this:

ii.a) At most one of the persons, Stav or David is in cabal or none of them is in cabal.

ii.b) Either Stav is not in cabal or David is not in cabal or both are not in cabal. But it can never be that both are in cabal.

(iii)

If either Martyna or Patrice are in cabal, then everyone is in cabal.

(iv)

If Stav is in cabal, then David is in cabal too

(v)

If Darren is in cabal, then Martyna is also in cabal.

(vi)

vi.a) If Oscar or Nick are in cabal, then Tom is not in cabal.

vi.b) If Tom is in cabal, then neither Oscar nor Nick are in cabal.

The reason why both sentences are equivalent is because of

*contrapositive*. Contrapositive says:

We can easily prove this with the truth table. The truth table for the implication looks like this:

true | true | true |

true | false | false |

false | true | true |

false | false | true |

An implication is true when the if part, i.e. P, is false or the then part, i.e. Q, is true. Otherwise it is false. Consequently the contrapositive of the implication looks like this:

false | false | true |

true | false | false |

false | true | true |

true | true | true |

If you compare the results of both tables, then you'll see that both are identical.

So, in this case these two expressions are equivalent:

(vi.a)

or

(vi.b)

The last proposition ((vi.b)) was derived by

**which says:**

__De Morgan's law__(vii)

vii.a) If Oscar or David are in cabal, then Marten is not in cabal.

vii.b) If Marten is in cabal, then neither Oscar nor David are in cabal.

Again, vii.b holds because it is contrapositive to vii.a) just like in the previous example.

Task 2)

So, in order to find who is in the cabal, we must must find a set of people, so that the propositions above (i)-(vii) are valid.

The proposition (iii) says, if Martyna and Patrice are in the cabal, then everyone is in cabal. But we know from (ii) already that either Stav or David is in cabal or none of them - but never both of them. So, the case that all of the people are in cabal can't be true. Therefore we know that:

(viii) neither Martyna nor Patrice are in the cabal.

The proposition (iv) says if Stave is in cabal, then David must also be in cabal. But (ii) already states that both can never be in cabal at the same time. Therefore we know that Stav is not in the cabal.

Since we know that Martyna is not in cabal (viii), it means that - because of (v) -

(ix) Darren isn't in the cabal either.

At this point, I think it should be clear why. Now it's going to be difficult. At least it was for me. I spent several hours to figure it out. What I did was to manually test the last two propositions:

According to (vi.b) if Tom is in cabal, then Oscar and Nick are not in the cabal. This proposition is either true or false.

If it is true - and if we compare it with the original set - then it means that only David, Marten and Tom are in the cabal. Now, let's have a look at the last proposition and we see that it says (vii.b) if Marten is in the cabal, then Oscar and David are not in the cabal. But we just assumed that David, Marten and Tom are in the cabal - if Tom is in the cabal. Since we know from (vii.b) that Marten and David can't be in the cabal at the same time, it means that Tom is not in the cabal. Therefore

(x) either Oscar or Nick are in the cabal. Or both are in the cabal. And Tom is not in the cabal.

Looking at (vii), there are two cases: Either Marten is in the cabal or Marten is not in the cabal. Let's assume that Marten is in the cabal. That would mean that neither Oscar nor David are in the cabal. That would mean that the cabal only exists of Nick and Marten. But our proposition (i) says that at least 3 different people are in the cabal. Therfore

(xi) Marten is not in the cabal.

This leaves us with these three candidates:

**Oscar, David, and Nick**. Since (i) says the cabal contains at least 3 people. Since we already have 3 people and can't exclude any more members, we can now stop and know that these three person are member of the cabal, because.

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