Chapter 1: Sets
Q1. The set of intelligent students in a class is:
Solution For a collection to be a set, it must be well-defined. "Intelligent" is subjective, so it is not well-defined.
Q2. If \(A = \{1, 2, 3, 4\}\), \(B = \{2, 4, 6, 8, 10\}\) and \(U = \{1, 2, ..., 10\}\), then \((A \cup B)'\):
Solution \(A \cup B = \{1, 2, 3, 4, 6, 8, 10\}\). The complement \((A \cup B)' = U - (A \cup B) = \{5, 7, 9\}\).
Q3. If \(A = \{1, 2, 3, 4\}\), \(B = \{2, 3, 5, 6\}\) and \(C = \{3, 4, 6, 7\}\), then \(A - (B \cap C)\):
Solution \(B \cap C = \{3, 6\}\). So, \(A - \{3, 6\} = \{1, 2, 4\}\).
Q4. Which of the following is correct?
Solution De Morgan's Law states \((A \cup B)' = A' \cap B'\). Option (b) is incorrect notation in the original image, so (d) is the correct choice.
Q5. The number of proper subsets of \(\{a, b, c\}\) is:
Solution Total subsets = \(2^3 = 8\). Proper subsets = \(2^n - 1 = 8 - 1 = 7\).
Q6. Which one is different from the others?
(i) empty set (ii) void set (iii) zero set (iv) null set
(i) empty set (ii) void set (iii) zero set (iv) null set
Solution Empty, void, and null set are synonyms for \(\phi\). A "zero set" implies \(\{0\}\), which is not empty.
Q7. If \(A=\{1,2,3,4\}\) and \(B=\{3,4,5,6\}\), then \(A - B = ?\)
Solution \(A - B\) is elements in A but not B. Here, \(1, 2\) are in A but not B.
Q8. If \(A = \{x, y\}\) then the power set of A is:
Solution Power set contains all subsets: \(\phi, \{x\}, \{y\}, \{x, y\}\).
Q9. The set \(\{x : x \text{ is an even prime number} \}\) can be written as:
Solution 2 is the only even prime number.
Q10. Given \(A=\{1,3,5\}\), \(B=\{2,4,6\}\) and \(C=\{0,2,4,6,8\}\). Which is universal set?
Solution The universal set must contain all elements of A, B, and C. Option (c) contains all of them.
Q11. If \(A \cup B \neq \phi\), then \(n(A \cup B) = ?\)
Solution Standard formula for union of sets.
Q12. Which of the following collections are sets?
Solution "Best", "rich", and "dangerous" are subjective. "Days of a week" is well-defined.
Q13. Which of the following properties is associative law?
Solution \((A \cup B) \cup C = A \cup (B \cup C)\) is the Associative Law.
Q14. Let \(V=\{a,e,i,o,u\}\) and \(B=\{a,i,k,u\}\). Value of \(V-B\) and \(B-V\) are:
Solution \(V-B = \{e, o\}\) (in V not B). \(B-V = \{k\}\) (in B not V).
Q15. Let \(A=\{a,b\}, B=\{a,b,c\}\). What is \(A \cup B\)?
Solution Union contains all elements: \(\{a, b, c\}\).
Q16. If A and B are finite sets, then \(n(A-B) = ?\)
Solution Elements in A minus those common to B.
Q17. If \(\phi\) denotes the empty set, then which is correct?
Solution \(\{\phi\}\) is a set containing the empty set as an element. Thus \(\phi \in \{\phi\}\).
Q18. Which one of the following is an infinite set?
Solution The set of prime numbers is infinite. Other physical examples are finite.
Q19. Let \(A=\{x : x \text{ is mul of } 3\}\), \(B=\{x : x \text{ is mul of } 5\}\). Then \(A \cap B\) is:
Solution Multiples of both 3 and 5 are multiples of 15.
Q20. The set \(A = \{x : x \in R, x^2 = 16 \text{ and } 2x = 6\}\) equals:
Solution \(x^2=16 \Rightarrow x=\pm 4\). \(2x=6 \Rightarrow x=3\). No number satisfies both. Empty set.