1. Questions, Options, Correct Answers, and Reasons
(i) The set builder form of the set \( \left\{1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots \right\}\) is:
(a) \(\{x \, | \, x = \frac{1}{n}, \, n \in W\}\)
(b) \(\{x \, | \, x = \frac{1}{2n + 1}, \, n \in W\}\)
(c) \(\{x \, | \, x = \frac{1}{n + 1}, \, n \in W\}\)
(d) \(\{x \, | \, x = 2n + 1, \, n \in W\}\)
Correct Answer: (b) \(\{x \, | \, x = \frac{1}{2n + 1}, \, n \in W\}\)
Reason: The denominators of the given fractions are odd numbers, which can be expressed as \(2n + 1\).
(ii) If \(A = \{\}\), then \(P(A)\) is:
(a) \(\{\}\)
(b) \(\{1\}\)
(c) \(\{\{\}\}\)
(d) \(\phi\)
Correct Answer: (c) \(\{\{\}\}\)
Reason: The power set of an empty set contains only one element: the empty set itself.
(iii) If \(U = \{1, 2, 3, 4, 5\}\), \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(U – (A \cap B)\) is:
(a) \(\{1, 2, 4, 5\}\)
(b) \(\{2, 3\}\)
(c) \(\{1, 3, 4, 5\}\)
(d) \(\{1, 2, 3\}\)
Correct Answer: (a) \(\{1, 2, 4, 5\}\)
Reason: \(A \cap B = \{3\}\), so \(U – (A \cap B) = \{1, 2, 4, 5\}\).
(iv) If \(A\) and \(B\) are overlapping sets, then \(n(A – B)\) is equal to:
(a) \(n(A)\)
(b) \(n(B)\)
(c) \(A \cap B\)
(d) \(n(A) – n(A \cap B)\)
Correct Answer: (d) \(n(A) – n(A \cap B)\)
Reason: Elements in \(A – B\) are those in \(A\) but not in \(B\), which is \(n(A) – n(A \cap B)\).
(v) If \(A \subseteq B\) and \(B – A \neq \phi\), then \(n(B – A)\) is equal to:
(a) \(0\)
(b) \(n(B)\)
(c) \(n(A)\)
(d) \(n(B) – n(A)\)
Correct Answer: (d) \(n(B) – n(A)\)
Reason: The number of elements in \(B – A\) is the total elements in \(B\) minus the elements in \(A\).
(vi) If \(n(A \cup B) = 50\), \(n(A) = 30\) and \(n(B) = 35\), then \(n(A \cap B) = \):
(a) \(23\)
(b) \(15\)
(c) \(9\)
(d) \(40\)
Correct Answer: (b) \(15\)
Reason: \(n(A \cup B) = n(A) + n(B) – n(A \cap B)\), so \(50 = 30 + 35 – n(A \cap B)\), solving gives \(n(A \cap B) = 15\).
(vii) If \(A = \{1, 2, 3, 4\}\) and \(B = \{x, y, z\}\), then Cartesian product of \(A\) and \(B\) contains exactly ______ elements.
(a) \(13\)
(b) \(12\)
(c) \(10\)
(d) \(6\)
Correct Answer: (b) \(12\)
Reason: The Cartesian product contains \(4 \times 3 = 12\) elements.
(viii) If \(f(x) = x^2 – 3x + 2\), then the value of \(f(a + 1)\) is equal to:
(a) \(a + 1\)
(b) \(a^2 + 1\)
(c) \(a^2 + 2a + 1\)
(d) \(a^2 – a\)
Correct Answer: (c) \(a^2 + 2a + 1\)
Reason: Substitute \(x = a + 1\) in \(f(x)\):
\[f(a + 1) = (a + 1)^2 – 3(a + 1) + 2 = a^2 + 2a + 1 – 3a – 3 + 2 = a^2 – a\]
(ix) Given \(f(x) = 3x + 1\), if \(f(x) = 28\), then the value of \(x\) is:
(a) \(9\)
(b) \(27\)
(c) \(3\)
(d) \(18\)
Correct Answer: (a) \(9\)
Reason: Solve \(3x + 1 = 28\):
\[3x = 27 \]
\[ x = 9\]
(x) Let \(A = \{1, 2, 3\}\) and \(B = \{a, b\}\) be two non-empty sets and \(f : A \to B\) be a function defined as \(f = \{(1, a), (2, b), (3, b)\}\). Which of the following is true?
(a) \(f\) is injective
(b) \(f\) is surjective
(c) \(f\) is bijective
(d) \(f\) is into only
Correct Answer: (b) \(f\) is surjective
Reason: Every element in \(B\) is mapped to by at least one element in \(A\), but \(f\) is not injective as \(b\) is mapped by both \(2\) and \(3\).
2. Write each of the following sets in tabular form:
(i) \(\{x \, | \, x = 2n, \, n \in \mathbb{N}\}\)
\[\{2, 4, 6, 8, 10, \dots\}\]
(ii) \(\{x \, | \, x = 2m + 1, \, m \in \mathbb{N}\}\)
\[\{3, 5, 7, 9, 11, \dots\}\]
(iii) \(\{x \, | \, x = 11n, \, n \in \mathbb{W} \, \land \, n < 11\}\)
\[\{0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110\}\]
(iv) \(\{x \, | \, x \in E \, \land \, 4 < x < 6\}\)
\[\{5\}\]
(v) \(\{x \, | \, x \in \mathbb{O} \, \land \, 5 \leq x < 7\}\)
\[\{5\}\]
(vi) \(\{x \, | \, x \in \mathbb{Q} \, \land \, x^2 = 2\}\)
\[\{\}\]
No rational number satisfies \(x^2 = 2\).
(vii) \(\{x \, | \, x \in \mathbb{Q} \, \land \, x = -x\}\)
\[\{0\}\]
(viii) \(\{x \, | \, x \in \mathbb{R} \, \land \, x \notin \mathbb{Q}\}\)
\[\text{Set of all irrational numbers. Example: } \{\pi, \sqrt{2}, \sqrt{3}, e, \dots\}\]
3. Let \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{2, 4, 6, 8, 10\}\), \(B = \{1, 2, 3, 4, 5\}\), and \(C = \{1, 3, 5, 7, 9\}\). List the members of each of the following sets:
(i) \(A’ = U – A\)
Given:
\[U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, \, A = \{2, 4, 6, 8, 10\}\]
\[A’ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} – \{2, 4, 6, 8, 10\}\]
\[A’ = \{1, 3, 5, 7, 9\}\]
(ii) \(B’ = U – B\)
Given:
\[U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, \, B = \{1, 2, 3, 4, 5\}\]
\[B’ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} – \{1, 2, 3, 4, 5\}\]
\[B’ = \{6, 7, 8, 9, 10\}\]
(iii) \(A \cup B\)
Given:
\[A = \{2, 4, 6, 8, 10\}, \, B = \{1, 2, 3, 4, 5\}\]
\[A \cup B = \{2, 4, 6, 8, 10\} \cup \{1, 2, 3, 4, 5\}\]
\[A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\}\]
(iv) \(A – B\)
Given:
\[A = \{2, 4, 6, 8, 10\}, \, B = \{1, 2, 3, 4, 5\}\]
\[A – B = \{2, 4, 6, 8, 10\} – \{1, 2, 3, 4, 5\}\]
\[A – B = \{6, 8, 10\}\]
(v) \(A \cap C\)
Given:
\[A = \{2, 4, 6, 8, 10\}, \, C = \{1, 3, 5, 7, 9\}\]
\[A \cap C = \{2, 4, 6, 8, 10\} \cap \{1, 3, 5, 7, 9\}\]
\[A \cap C = \{\}\]
(vi) \(A’ \cup C’\)
Given:
\[A’ = U – A, \, C’ = U – C\]
\[A’ = \{1, 3, 5, 7, 9\}, \, C’ = \{2, 4, 6, 8, 10\}\]
\[A’ \cup C’ = \{1, 3, 5, 7, 9\} \cup \{2, 4, 6, 8, 10\}\]
\[A’ \cup C’ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\]
(vii) \(A’ \cap C’\)
Given:
\[A’ = \{1, 3, 5, 7, 9\}, \, C’ = \{2, 4, 6, 8, 10\}\]
\[A’ \cap C’ = \{1, 3, 5, 7, 9\} \cap \{2, 4, 6, 8, 10\}\]
\[A’ \cap C’ = \{\}\]
(viii) \(U’\)
Given:
\[U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\]
\[U’ = \phi\]
\[U’ = \phi\]
6. Verify the properties for the sets \(A\), \(B\), and \(C\):
(i) Associativity of Union: \((A \cup B) \cup C = A \cup (B \cup C)\):
Given:
\[A = \{1, 2, 3, 4\}, \, B = \{3, 4, 5, 6, 7, 8\}, \, C = \{5, 6, 7, 9, 10\}\]
Find \(A \cup B\):
\[A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\]
Find \((A \cup B) \cup C\):
\[(A \cup B) \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\]
Find \(B \cup C\):
\[B \cup C = \{3, 4, 5, 6, 7, 8, 9, 10\}\]
Find \(A \cup (B \cup C)\):
\[A \cup (B \cup C) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\]
Verify:
\[(A \cup B) \cup C = A \cup (B \cup C)\]
(ii) Associativity of Intersection: \((A \cap B) \cap C = A \cap (B \cap C)\):
Find \(A \cap B\):
\[A \cap B = \{3, 4\}\]
Find \((A \cap B) \cap C\):
\[(A \cap B) \cap C = \{\}\]
Find \(B \cap C\):
\[B \cap C = \{5, 6, 7\}\]
Find \(A \cap (B \cap C)\):
\[A \cap (B \cap C) = \{\}\]
Verify:
\[(A \cap B) \cap C = A \cap (B \cap C)\]
(iii) Distributivity of Union over Intersection: \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\):
Find \(B \cap C\):
\[B \cap C = \{5, 6, 7\}\]
Find \(A \cup (B \cap C)\):
\[A \cup (B \cap C) = \{1, 2, 3, 4, 5, 6, 7\}\]
Find \(A \cup B\):
\[A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\]
Find \(A \cup C\):
\[A \cup C = \{1, 2, 3, 4, 5, 6, 7, 9, 10\}\]
Find \((A \cup B) \cap (A \cup C)\):
\[(A \cup B) \cap (A \cup C) = \{1, 2, 3, 4, 5, 6, 7\}\]
Verify:
\[A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\]
(iv) Distributivity of Intersection over Union: \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\):
Find \(B \cup C\):
\[B \cup C = \{3, 4, 5, 6, 7, 8, 9, 10\}\]
Find \(A \cap (B \cup C)\):
\[A \cap (B \cup C) = \{3, 4\}\]
Find \(A \cap B\):
\[A \cap B = \{3, 4\}\]
Find \(A \cap C\):
\[A \cap C = \{\}\]
Find \((A \cap B) \cup (A \cap C)\):
\[(A \cap B) \cup (A \cap C) = \{3, 4\}\]
Verify:
\[A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\]
7. Verify De Morgan’s Laws for the given sets:
Given:
\[U = \{1, 2, 3, \dots, 20\}, \, A = \{2, 4, 6, \dots, 20\}, \, B = \{1, 3, 5, \dots, 19\}\]
(i) \((A \cup B)^c = A^c \cap B^c\):
Find \(A \cup B\):
\[A \cup B = \{1, 2, 3, 4, 5, \dots, 20\}\]
Find \((A \cup B)^c\):
\[(A \cup B)^c = \phi\]
Find \(A^c\):
\[A^c = \phi\]
Find \(B^c\):
\[B^c = \phi\]
Find \(A^c \cap B^c\):
\[A^c \cap B^c = \phi\]
Verify:
\[(A \cup B)^c = A^c \cap B^c\]
(ii) \((A \cap B)^c = A^c \cup B^c\):
Find \(A \cap B\):
\[A \cap B = \phi\]
Find \((A \cap B)^c\):
\[(A \cap B)^c = \{1, 2, 3, \dots, 20\}\]
Find \(A^c\):
\[A^c = \phi\]
Find \(B^c\):
\[B^c = \phi\]
Find \(A^c \cup B^c\):
\[A^c \cup B^c = \{1, 2, 3, \dots, 20\}\]
Verify:
\[(A \cap B)^c = A^c \cup B^c\]
8. Find \(P \cap Q\):
Given:
\[P = \{x \, | \, x = 5m, \, m \in \mathbb{N}\}, \, Q = \{x \, | \, x = 2m, \, m \in \mathbb{N}\}\]
List some elements of \(P\):
\[P = \{5, 10, 15, 20, 25, \dots\}\]
List some elements of \(Q\):
\[Q = \{2, 4, 6, 8, 10, 12, \dots\}\]
Find common elements:
\[P \cap Q = \{10, 20, 30, \dots\}\]
Generalize:
\[P \cap Q = \{x \, | \, x = 10m, \, m \in \mathbb{N}\}\]
9. Verify properties of union and intersection:
(i) \(A \cap (A \cup B) = A \cup (A \cap B)\):
Find \(A \cap (A \cup B)\):
\[A \cap (A \cup B) = A\]
Find \(A \cup (A \cap B)\):
\[A \cup (A \cap B) = A\]
Verify:
\[A \cap (A \cup B) = A \cup (A \cap B)\]
(ii) \(A \cup (A \cap B) = A \cap (A \cup B)\):
Find \(A \cup (A \cap B)\):
\[A \cup (A \cap B) = A\]
Find \(A \cap (A \cup B)\):
\[A \cap (A \cup B) = A\]
Verify:
\[A \cup (A \cap B) = A \cap (A \cup B)\]
10. If \(g(x) = 7x – 2\) and \(s(x) = 8x^2 – 3\), find the following:
(i) \(g(0)\):
\[g(x) = 7x – 2\]
Substitute \(x = 0\):
\[g(0) = 7(0) – 2 = -2\]
(ii) \(g(-1)\):
\[g(x) = 7x – 2\]
Substitute \(x = -1\):
\[g(-1) = 7(-1) – 2 = -7 – 2 = -9\]
(iii) \(g\left(-\frac{5}{3}\right)\):
\[g(x) = 7x – 2\]
Substitute \(x = -\frac{5}{3}\):
\[g\left(-\frac{5}{3}\right) = 7\left(-\frac{5}{3}\right) – 2 = -\frac{35}{3} – 2 = -\frac{35}{3} – \frac{6}{3} = -\frac{41}{3}\]
(iv) \(s(1)\):
\[s(x) = 8x^2 – 3\]
Substitute \(x = 1\):
\[s(1) = 8(1)^2 – 3 = 8 – 3 = 5\]
(v) \(s(-9)\):
\[s(x) = 8x^2 – 3\]
Substitute \(x = -9\):
\[s(-9) = 8(-9)^2 – 3 = 8(81) – 3 = 648 – 3 = 645\]
(vi) \(s\left(\frac{7}{2}\right)\):
\[s(x) = 8x^2 – 3\]
Substitute \(x = \frac{7}{2}\):
\[s\left(\frac{7}{2}\right) = 8\left(\frac{7}{2}\right)^2 – 3 = 8\left(\frac{49}{4}\right) – 3 = \frac{392}{4} – 3 = 98 – 3 = 95\]
11. Given \(f(x) = ax + b\), \(f(-2) = 3\), and \(f(4) = 10\), find \(a\) and \(b\):
Form equations using given conditions:
\[f(x) = ax + b\]
Substitute \(x = -2\) and \(f(-2) = 3\):
\[3 = -2a + b \quad \text{(Equation 1)}\]
Substitute \(x = 4\) and \(f(4) = 10\):
\[10 = 4a + b \quad \text{(Equation 2)}\]
Solve the system of equations:
Subtract Equation 1 from Equation 2:
\[(10 – 3) = (4a + b) – (-2a + b)\]
\[7 = 6a \]
\[ a = \frac{7}{6}\]
Substitute \(a = \frac{7}{6}\) into Equation 1:
\[3 = -2\left(\frac{7}{6}\right) + b\]
\[3 = -\frac{14}{6} + b\]
\[b = 3 + \frac{14}{6} = \frac{18}{6} + \frac{14}{6} = \frac{32}{6} = \frac{16}{3}\]
Final Answer:
\[a = \frac{7}{6}, \, b = \frac{16}{3}\]
12. Given \(k(x) = 7x – 5\), find \(x\) if \(k(x) = 100\):
Set \(k(x) = 100\):
\[7x – 5 = 100\]
\[7x = 100 + 5 = 105\]
\[x = \frac{105}{7} = 15\]
Final Answer:
\[x = 15\]
13. Given \(g(x) = mx^2 + n\), \(g(4) = 20\), and \(g(0) = 5\), find \(m\) and \(n\):
Form equations using the given conditions:
Substitute \(x = 4\) and \(g(4) = 20\):
\[20 = m(4)^2 + n\]
\[20 = 16m + n \quad \text{(Equation 1)}\]
Substitute \(x = 0\) and \(g(0) = 5\):
\[5 = m(0)^2 + n\]
\[n = 5 \quad \text{(Equation 2)}\]
Solve for \(m\):
Substitute \(n = 5\) into Equation 1:
\[20 = 16m + 5\]
\[16m = 20 – 5 = 15\]
\[m = \frac{15}{16}\]
Final Answer:
\[m = \frac{15}{16}, \, n = 5\]
