SOLUTION: 3. Happy Harold’s Autos collected data on cars sold in October. 43 Total number of cars sold 23 Red cars 25 Cars with Air Conditioning 29 Cars with Heated Seats 17 Red car

You are given the universal set U of 43 cars and 3 its basic subsets R, A and H 
(see the table below).

    U    43    total cars

    R    23    Red cars

    A    25    cars with Air conditioning

    H    29    cars with Heated seats


Also, you are given info about their in-pair intersections and about their triple intersection.


    RA   17    Red cars with Air conditioning

    RH   15    Red cars with Heated seats

    AH   16    cars with Air conditioning and Heated seats

    RAH  10    Red cars with Air conditioning and Heated seats



Having this info well organized, you can easily answer all questions (b), (c) and (d) 
without drawing Venn diagram.



(b)  The set {Red cars had neither Air Conditioning nor Heated Seats} is  R \ ((RA U RH) \ RAH)   

     and their number is  23 - (17 + 15 - 10)  = 1.    ANSWER                                     



(c)  The set {cars had Air Conditioning were not red} is  A \ RA,

     and their number is  25 - 17 = 8.     ANSWER



(d)  The set {cars were not red and had neither Air Conditioning nor Heated Seats} is  U \ (R U A U H).
     
     So, calculate the number of cars in the union (R U A U H}  first.

     For it, use the inclusion-exclusion princuple/(formula)

         n(R U A U H) = n(R) + n(A) + n(H) - n(RA) - n(RH) - n(AH) + n(RAH) = 

                      =  23  +  25  +  29  -  17   -  15   -  16   +  10    = 39.


     Now the last step gives the answer to question (d) :  

        the number of cars were not red and had neither Air Conditioning nor Heated Seats = 43 - 39 = 4.    ANSWER

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