Theorems on Sets, venn diagrams

Question 11

Question:  

Answer:   




From (i) and (ii),

Similarly,

Question 12

Question:  

Answer:   


(B^l contains all the elements of A and some other elements also.)

Question 13

Question:  

Answer:   

Question 14

Question:  

Answer:    I. Proof:





II. Proof:




.........(i)




........ (ii)
From (i) and (ii),

Question 15

Question:  

Answer:   
Let





... (i)
Let,





... (ii)
From (i) and (ii), we get

Question 16

Question:  

Answer:   
Let us assume,


and let


But x B and x B cannot hold good simulataneously. This is a contradiction. Hence our assumption is wrong and the theorem is true.

Question 17

Question:  

Answer:    We shall prove this in two parts,

Question 18

Question:  

Answer:   
Thus verified.

Question 19

Question:   Each person in a group of 50 can speak either English or Hindi or both. If 35 persons can speak English and 25 can speak both; find the number of those who speak Hindi only.

Answer:    Let
n (E) = number of persons who can speak English
n (H) = number of persons who can speak Hindi

To find:

Question 20

Question:   There are 64 persons in a group. Each of them eats either apple or plantain or both. Those who eat apples are 36 in number. The number of those who eat both apples and plantain is 21. How many eat plantains only?

Answer:    Let n(A) = number of persons who eat apples
n(P) = number of persons who eat plantains