As given in question, we have three friends (J, S, K) & Everyone has two possibilities either Present or Absent.
So, using this data we can make total $2^3=8$ combinations.
by translating given english statements into logical statements. we have three logical statements,
(J$\rightarrow\neg$S), (S$\rightarrow$K), ($\neg$J$\rightarrow\neg$K) .
As we know that (p$\rightarrow$q) is false only for p is True & q is False. we apply this rule on above three logical statements,
Now result is,
J is True $\rightarrow$ S is True -----(1) |
S is True $\rightarrow$ K is False ----(2) |
J is False $\rightarrow$ K is True ----(3) |
by using this result we can easily eliminate false combination.
Now, we make table for 8 combinations and eliminate false combination. Suppose, T=Present & F=Absent.
J |
S |
K |
|
F |
F |
F |
|
F |
F |
T |
false(using 3rd logical statement). |
F |
T |
F |
false(using 2nd logical statement). |
F |
T |
T |
false(using 3rd logical statement). |
T |
F |
F |
|
T |
F |
T |
|
T |
T |
F |
false(using 1st or 2nd logical statement). |
T |
T |
T |
false(using 1st logical statement). |
Remaining valid combinations for (J, S, K) are (F, F, F), (T, F, F), (T, F, T).
Hence, we can invite these three friends in above valid combinations, so as not to make someone unhappy.