Chapter 1 -Sets

Show that \(A \cup B = A \cap B\) implies \(A = B\).

Is the set \(T = \{x | \frac{x+5}{x-7} - 5 = \frac{4x-40}{13-x}\}\) an empty set? Justify your answer.

For all sets \(A, B\) and \(C\), show that \((A - B) \cap (A - C) = A - (B \cup C)\).

Let \(A, B\) and \(C\) be sets. Then show that \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed
(i) in English and Mathematics but not in Science
(ii) in Mathematics and Science but not in English
(iii) in Mathematics only
(iv) in more than one subject only

In a survey of 60 people, 25 read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

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