denotes the number of men who got medal in all the three. Thus, d = n ( F ∩ B ∩ C ) = and a + d + b + d + c + d = Therefore a + b + c = , which is the number of people who got medals in exactly two of the three sports. Miscellaneous Exercise on Chapter . Decide, among the following sets, which sets are subsets of one and another: A = { x : x ∈ R and x satisfy x – x + = }, B = { , , }, C = { , , , , .
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. If x ∈ A and A ∈ B , then x ∈ B 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 .
Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C. . Show that the following four conditions are equivalent : (i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A .
Show that if A ⊂ B, then C – B ⊂ C – A. . Assume that P ( A ) = P ( B ). Show that A = B .
Is it true that for any sets A and B, P ( A ) ∪ P ( B ) =