Q7. True or false: (i) {a} ⊂ {a, b, c}; (ii) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36} (NCERT Exercise 1.3, Q2 (iv), (vi)).
(i) True. (ii) True: the first set is {2, 4}, and both 2 and 4 divide 36.
Q8. Fill in ⊂ or ⊄: (i) {x : x is a student of Class XI of your school} … {x : x is a student of your school}; (ii) {x : x is a circle in the plane} … {x : x is a circle in the same plane with radius 1 unit}; (iii) {x : x is a triangle in a plane} … {x : x is a rectangle in the plane} (NCERT Exercise 1.3, Q1 (iii)–(v)).
(i) ⊂. (ii) ⊄: a circle of radius 2 is in the first set but not the second. (iii) ⊄: a triangle is not a rectangle.
Q9. Write [−10, −2) and (6, 12) in set-builder form, and give the length of each.
{x : x ∈ R, −10 ≤ x < −2}, length 8; {x : x ∈ R, 6 < x < 12}, length 6.
Q10. Decide which are subsets of one another: A = {x : x ∈ R and x2 − 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, …}, D = {6} (NCERT Miscellaneous Exercise, Q1).
x2 − 8x + 12 = (x − 2)(x − 6), so A = {2, 6}. Then A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B and D ⊂ C.
Q11. Is (−3, 5) a subset of [−7, 9]? Is [−7, 9] a subset of (−3, 5)?
Yes: every x with −3 < x < 5 also satisfies −7 ≤ x ≤ 9 (NCERT page 11). No: for example −5 is in [−7, 9] but not in (−3, 5).
Q12. True or false: (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 x ∉ B, then x ∉ A (NCERT Miscellaneous Exercise, Q2 (i), (ii), (vi)).
(i) False: A = {1}, B = {{1}} gives 1 ∈ A and A ∈ B, but 1 ∉ B. (ii) False: A = {1}, B = {1, 2}, C = {{1, 2}} gives A ⊂ B and B ∈ C, but A ∉ C. (iii) True: if x were in A, A ⊂ B would put x in B.