George, John, Ivy, Paul, and Kurt attend the annual general assembly

Question

Help me find every scenario that satisfies all of the premises.

Suppose that the president of a club has the following information about the attendance of George, John, Ivy, Paul, and Kurt to the annual general assembly of school clubs:

1. Exactly two of George, John, and Paul attended.
2. If Kurt attended, then John attended.
3. Either both John and Kurt attended, or John did not attend.
4. If John attended, then neither George nor Ivy attended.
5. At least one of Paul and George attended.
6. Paul attended if and only if Kurt was absent.
7. If George attended then so did Ivy.

Determine the possible attendance scenarios.

Answer 1

If John attended, Kurt attended (third sentence), George was absent and Ivy was absent (fourth sentence), so Paul attended (first sentence). But this contradicts the sixth sentence.

Thus, John did not attend. Then George and Paul attended (first sentence), and Kurt was absent (sixth sentence), and Ivy attended (last sentence). This scenario satisfies all the other conditions.

Answer 2

"Exactly two of George, John, and Paul attended"

"If John attended, then neither George nor Ivy attended."

So it can't be George and John. So it must be Paul and either George or John. But definitely Paul.

" Paul attended if and only if Kurt was absent."

So Kurt was absent.

" Either both John and Kurt attended, or John did not attend. "

Kurt was absent so John was absent.

So it must be Paul and either George or John. But definitely Paul.

But John was absent, so Paul and George attended.

So.... what have we got?

"Exactly two of George, John, and Paul attended." Fine, it was Paul and George.

"If Kurt attended, the John attended." He didn't.

" Either both John and Kurt attended, or John did not attend." Neither attended.

" If John attended, then neither George nor Ivy attended." Irrelevant as John did not attend.

"At least one of Paul and George attended." Actually, both of them did.

"Paul attended if and only if Kurt was absent." Paul did, Kurt was.

" If George attended then so did Ivy." George attended so Ivy did.

Sooo...

George: atteneded

John: absent

Ivy:attended

Paul: attended

Kurt: absent.

Answer 3

We introduce the following atomic propositions

• $p$: George attended the annual general assembly
• $q$: John attended the annual general assembly
• $r$: Ivy attended the annual general assembly
• $s$: Paul attended the annual general assembly
• $t$: Kurt attended the annual general assembly

Translating the seven constraints,

1. Exactly two of George, John, and Paul attended.

$$\Phi_1 := (p \land q \land \neg s) \lor (p \land \neg q \land s) \lor (\neg p \land q \land s)$$

2. If Kurt attended, then John attended.

$$\Phi_2 := t \to q$$

3. Either both John and Kurt attended, or John did not attend.

$$\Phi_3 := (q \land t) \lor \neg q$$

4. If John attended, then neither George nor Ivy attended.

$$\Phi_4 := q \to (\neg p \land \neg r)$$

5. At least one of Paul and George attended.

$$\Phi_5 := (p \land s) \lor (\neg p \land s) \lor (p \land \neg s)$$

6. Paul attended if and only if Kurt was absent.

$$\Phi_6 := (s \to \neg t) \land (\neg t \to s)$$

7. If George attended then so did Ivy.

$$\Phi_7 := p \to r$$

Let

$$\Phi := \bigwedge_{i=1}^7 \Phi_i$$

Using SymPy Live,

>>> from sympy import logic
>>> p, q, r, s, t = symbols('p q r s t')
>>> Phi1 = (p & q & Not(s)) | (p & Not(q) & s) | (Not(p) & q & s)
>>> Phi2 = t >> q
>>> Phi3 = (q & t) | Not(q)
>>> Phi4 = q >> (Not(p) & Not(r))
>>> Phi5 = (p & s) | (Not(p) & s) | (p & Not(s))
>>> Phi6 = (s >> Not(t)) & (Not(t) >> s)
>>> Phi7 = p >> r
>>> Phi = Phi1 & Phi2 & Phi3 & Phi4 & Phi5 & Phi6 & Phi7
>>> Phi
And(Implies(p, r), Implies(q, And(Not(p), Not(r))), Implies(s, Not(t)), Implies(t, q), Implies(Not(t), s), Or(And(q, t), Not(q)), Or(And(p, s), And(p, Not(s)), And(s, Not(p))), Or(And(p, q, Not(s)), And(p, s, Not(q)), And(q, s, Not(p))))
>>> simplify(Phi)
And(p, r, s, Not(q), Not(t))

Hence,

$$\Phi \equiv p \land \neg q \land r \land s \land \neg t$$

We conclude that George, Ivy and Paul did attend the event. John and Kurt did not.