List of practice Questions

Make correct statements by filling in the symbols \( ⊂ \) or \(⊄\) in the blank spaces:
(i) {2, 3, 4} … {1, 2, 3, 4, 5}
(ii) {a, b, c} … {b, c, d}
(iii) {x: x is a student of Class XI of your school} … {x: x student of your school}
(iv) {x: x is a circle in the plane} … {x: x is a circle in the same plane with radius 1 unit}
(v) {x: x is a triangle in a plane}…{x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane}… {x: x is a triangle in the same plane} 
(vii) {x: x is an even natural number} … {x: x is an integer}

Examine whether the following statements are true or false:
(i) {a, b} \(⊄\) {b, c, a}
(ii) {a, e} \(⊂\) {x: x is a vowel in the English alphabet}
(iii) {1, 2, 3} \(⊂\) {1, 3, 5}
(iv) {a} \(⊂ \) {a. b, c}
(v) {a} \(∈\) (a, b, c)
(vi) {x: x is an even natural number less than 6} \(⊂\) {x: x is a natural number which divides 36}

Write down all the subsets of the following sets:
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) \(\phi\)

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
(i) \(A∪ B\)
(ii) \(A ∪ C\)
(iii) \(B ∪ C\)
(iv) \(B ∪ D\)
(v) \(A ∪ B ∪ C\)
(vi) \(A∪ B ∪ D\)
(vii) \(B ∪ C ∪ D\)

Decide, among the following sets, which sets are subsets of one and another:
A = {x: x \(\in\) R and x satisfy \( x^2 - 8x + 12 = 0\)},
B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.

Show that the following four conditions are equivalent: \((i) A ⊂ B (ii) A – B = \phi (iii) A ∪ B = B (iv) A ∩ B = A\)

Show that if \(A ⊂ B\), then \(C – B ⊂ C – A.\)

Show that for any sets A and B, \(A = (A ∩ B) ∪ (A – B)\) and \(A ∪ (B – A) = (A ∪ B)\)

Using properties of sets show that (i) \(A ∪ (A ∩ B) = A \) (ii) \(A ∩ (A ∪ B) = A.\)

Show that \(A ∩ B = A ∩ C\) need not imply B = C.

Let A and B be sets. If \(A ∩ X = B ∩ X = \phi\) 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)

Find sets A, B and C such that \(A ∩ B, B ∩ C\) and \(A ∩ C\) are non-empty sets and \(A ∩ B ∩ C = \phi.\)