Question

One hundred and twenty-five (125) aliens descended on a set of film as Extra Terrestrial Beings. 40 has two noses, 30 had three legs, 20 had four ears, 10 had two noses and three legs, 12 had three legs and four ears, 5 had two noses and four ears and 3 has all the three unusual feature. How many were there without any of these unusual features?
A) 5
B) 35
C) 80
D) 59

Answer 1

Hint: Venn diagrams consist of rectangles and closed curves usually circles. The universal set is represented by a rectangle and its subsets by a circle.

Complete step by step solution:
Step 1

 Total number of aliens = 125.
$\therefore \cup $: universal set is total number of aliens
$\therefore {\rm{n(}} \cup {\rm{)}}$: number of elements in universal set $ \cup $= 125 …… (1)

Step 2
Let A = set of two noses aliens
$\therefore $ n(A): number of elements in set A = 40 …… (2)

Step 3
Let B = set of three legged aliens
$\therefore $n(B): number of elements in set B = 30 …… (3)

Step 4
Let C = set of four ears aliens
$\therefore $n(C): number of elements in set A = 20 …… (4)

Step 5
${\rm{(A}} \cap {\rm{B):}}$Set of aliens having two noses and three legs
$\therefore {\rm{(A}} \cap {\rm{B):}}$number of element in set A intersection set B = 10 …… (5)

Step 6
${\rm{(B}} \cap {\rm{C):}}$Set of aliens having three legs and four ears
$\therefore {\rm{(B}} \cap {\rm{C):}}$number of element in set B intersection set C = 12 …… (6)

Step 7
${\rm{(A}} \cap {\rm{C):}}$Set of aliens having two noses and four ears
$\therefore {\rm{(A}} \cap {\rm{C):}}$number of element in set A intersection set C = 5 …… (7)

Step 8
${\rm{(A}} \cap {\rm{B}} \cap {\rm{C):}}$Set of aliens having all three unusual feature
$\therefore {\rm{(A}} \cap {\rm{B}} \cap {\rm{C):}}$number of element in set A intersection set B intersection set C = 3 …… (8)

Step 9
$({\rm{A}} \cup {\rm{B}} \cup {\rm{C}})$: Set of aliens having at least one of the unusual features.
$\therefore {\rm{n}}({\rm{A}} \cup {\rm{B}} \cup {\rm{C}}):$number of elements in set A union set B union set C:
${\rm{n(A}} \cup {\rm{B}} \cup {\rm{C) = n(A) + n(B) + n(C) - n(A}} \cap {\rm{B) - n(B}} \cap {\rm{C) - n(A}} \cap {\rm{C) + n(A}} \cap {\rm{B}} \cap {\rm{C)}}$
${\rm{n}}({\rm{A}} \cup {\rm{B}} \cup {\rm{C}}) = $40 + 30 + 20 – 10 – 12 – 5 +3 (from equation (1) to (8))
                         = 66 …… (9)

Step 10
$({\rm{A}} \cup {\rm{B}} \cup {\rm{C}})':$ Complement set
                      : Set of aliens having none of the above mentioned unusual features.
$({\rm{A}} \cup {\rm{B}} \cup {\rm{C}})' = \cup - ({\rm{A}} \cup {\rm{B}} \cup {\rm{C}})$
${\rm{n}}({\rm{A}} \cup {\rm{B}} \cup {\rm{C}})' = {\rm{n(}} \cup ) - {\rm{n}}({\rm{A}} \cup {\rm{B}} \cup {\rm{C}})$
                         = 125 – 66 (from equation (1) and (9))
                         = 59

Therefore, there are 59 such aliens having none of the unusual features. Correct option is D.

Note:
Most of the relationships between sets can be represented by Venn diagrams. For two sets:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$