1.12 Miscellaneous Exercise on Chapter 1

NCERT Class 11 Mathematics Textbook for blind and visually impaired students made Screen Readable by Professor T K Bansal.

Question 1.


Decide, among the following sets, which sets are subsets of one and another:


A = { x : x ∈ R and x satisfy x^2 − 8x + 12 = 0 },


B = { 2, 4, 6 },


C = { 2, 4, 6, 8, . . . },


D = { 6 }.

Answer 1.


A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C

Question 2.


In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.


(i) If x ∈ A and A ∈ B, then x ∈ B


(ii) If A ⊂ B and B ∈ C, then A ∈ C


(iii) If A ⊂ B and B ⊂ C, then A ⊂ C


(iv) If A ⊄ B and B ⊄ C, then A ⊄ C


(v) If x ∈ A and A ⊄ B, then x ∈ B


(vi) If A ⊂ B and x ∉ B, then x ∉ A

Answer 2.


(i) False


(ii) False


(iii) True


(iv) False


(v) False


(vi) True

Question 3.


Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.

Question 4.


Show that the following four conditions are equivalent :


(i) A ⊂ B


(ii) A − B = φ


(iii) A ∪ B = B


(iv) A ∩ B = A

Question 5.


Show that if A ⊂ B, then C − B ⊂ C − A.

Question 6.


Show that for any sets A and B,


A = ( A ∩ B ) ∪ ( A − B ) and A ∪ ( B − A ) = ( A ∪ B )

Question 7.


Using properties of sets, show that


(i) A ∪ ( A ∩ B ) = A


(ii) A ∩ ( A ∪ B ) = A.

Question 8.


Show that A ∩ B = A ∩ C need not imply B = C.

Question 9.


Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.


(Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )

Question 10.


Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.

Answer 10.


We may take


A = { 1, 2 },


B = { 1, 3 },


C = { 2 , 3 }