* 1
a.
    S = Sunny
    I = Ice cream
    B = Beer

    S => (I V B)

b. 
    F = Earth flat
    G = Earth a globe
    H = Global warming is hoax

    F => !G => H

c. 
    L = Attended lectures
    H = Homework done
    S = Studied for exam
    P = Passed maths

    (L A H A S ) <=> P

d. 
    S = You smell essential oils
    C = You will be cured

    N = Modern medicine doesnt work

    (S => C) <=> N

* 2

a. (P V !Q) <=> (Q => P)

    P   Q     P V !Q    Q => P
    1   1       1         1
    1   0       1         1
    0   1       0         0
    0   0       1         1

b. (P => (Q => R)) <=> ((P A Q) => R)

    R   Q   P    P => (Q => R)     (P A Q) => R
    1   1   1         1                 1
    1   1   0         1                 1
    1   0   0         1                 1
    0   0   0         1                 1
    0   0   1         1                 1
    0   1   0         1                 1
    1   0   1         1                 1
    0   1   1         0                 0

c. ((P => Q) V (P => R)) <=> (P => (Q V R))

    R   Q   P    (P => Q) V (P => R)     P => (Q V R)
    1   1   1             1                   1             
    1   1   0             1                   1
    1   0   0             1                   1
    0   0   0             1                   1
    0   0   1             0                   0
    0   1   0             1                   1
    1   0   1             1                   1
    0   1   1             1                   1

d. (!(P A !Q) V (R => 0)) <=> ((P A R) => Q)

    R   Q   P    !(P A !Q) V (R => 0)     (P A R) => Q
    1   1   1              1                    1    
    1   1   0              1                    1
    1   0   0              1                    1
    0   0   0              1                    1
    0   0   1              1                    1
    0   1   0              1                    1
    1   0   1              0                    0
    0   1   1              1                    1

* 3

a. (P A (Q V R)) and ((P A Q) V P)  = 2nd follows from 1st
    
    R   Q   P    P A (Q V R)    (P A Q) V P
    1   1   1         1              1    
    1   1   0         0              0          
    1   0   0         0              0          
    0   0   0         0              0          
    0   0   1         0              1          
    0   1   0         0              0          
    1   0   1         0              1          
    0   1   1         1              1          

b. (P V (Q A R)) and ((P V Q) A (P V R)) = 1st follows from 2nd

    R   Q   P    P V (Q A R)     (P V Q) A (P V R)
    1   1   1         1                  1                         
    1   1   0         1                  1      
    1   0   0         0                  0       
    0   0   0         0                  0       
    0   0   1         1                  0       
    0   1   0         0                  0       
    1   0   1         1                  1      
    0   1   1         1                  1       

c. (P => (Q => R)) and ((P => Q) => R) = 1st follows from 2nd
    
    R   Q   P    P => (Q => R)     (P => Q) => R
    1   1   1         1                  1                         
    1   1   0         1                  1       
    1   0   0         1                  1       
    0   0   0         1                  0       
    0   0   1         1                  1       
    0   1   0         1                  0       
    1   0   1         1                  1       
    0   1   1         0                  0       

d. (0 => (P A Q)) and (P V Q) = 1st follows from 2nd

    R   Q   P    0 => (P A Q)     P V Q
    1   1   1          1            1             
    1   1   0          1            1            
    1   0   0          1            0       
    0   0   0          1            0            
    0   0   1          1            1            
    0   1   0          1            1            
    1   0   1          1            1            
    0   1   1          1            1           


* 5

B = Bob says truth
A = Alice says truth
C = Carol says truth            
                                    
    B   A   C       !B => !C      !(!A A !C)          !A A !B     Contradiction
    1   1   1           1             1                  0              x      
    1   1   0           1             1                  0              *   
    1   0   0           1             0                  0              x
    0   0   0           1             0                  1              x
    0   0   1           0             1                  1              x
    0   1   0           1             1                  0              x
    1   0   1           1             1                  0              x
    0   1   1           0             1                  0              x

Bob = truth
Alice = truth
Carol = lie