Suppose \(A_1, A_2,\ldots,A_{30}\) are thirty sets each having 5 elements and \(B_1, B_2,\ldots,B_n\) are \(n\) sets each with 3 elements. Let \[ \bigcup_{i=1}^{30} A_i=\bigcup_{j=1}^{n} B_j=S \] and each element of \(S\) belongs to exactly 10 of the \(A_i\)’s and exactly 9 of the \(B_j\)’s, then \(n\) is equal to (a) 15 (b) 3 (c) 45 […]

In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics \(=100\) Physics \(=70\) Chemistry \(=40\) Mathematics and Physics \(=30\) Mathematics and Chemistry \(=28\) Physics and Chemistry \(=23\) Mathematics, Physics and Chemistry \(=18\) How many students have offered Mathematics alone? (a) 35 (b) 48 (c) 60

Two finite sets have \(m\) and \(n\) elements. The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second set. Then, the values of \(m\) and \(n\) are: (a) \(7,6\) (b) \(6,3\) (c) \(6,4\) (d) \(7,4\) Solution Number of elements

An investigator interviewed 100 students to determine the performance of three drinks: milk, coffee and tea. The investigator reported that \[ n(M\cap C\cap T)=10 \] \[ n(M\cap C)=20 \] \[ n(M\cap T)=25 \] \[ n(C\cap T)=20 \] Milk only \(=12\), Coffee only \(=5\), Tea only \(=8\) Solution Students taking only two drinks: \[ (M\cap C)\text{

If \(A \cap B = B\), then (a) \(A\subseteq B\) (b) \(B\subseteq A\) (c) \(A=\Phi\) (d) \(B=\Phi\) Solution \[ A\cap B=B \] This means every element of \(B\) belongs to \(A\). Therefore, \[ B\subseteq A \] Answer \[ \boxed{B\subseteq A} \] Correct option: (b) Next Question / Full Exercise

In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or bus is (a) 80% (b) 40% (c) 60% (d) 70% Solution Let \[ n(C)=20\% \] \[ n(B)=50\% \] \[ n(C\cap B)=10\% \] Using the formula, \[ n(C\cup

If \(A=\{x:x \text{ is a multiple of }3\}\) and \(B=\{x:x \text{ is a multiple of }5\}\), then \(A-B\) is (a) \(A\cap B\) (b) \(A\cap \overline{B}\) (c) \(\overline{A}\cap \overline{B}\) (d) \(\overline{A}\cap B\) Solution By definition, \[ A-B \] means the elements which belong to \(A\) but do not belong to \(B\). Therefore, \[ A-B=A\cap \overline{B} \]

If A and B are two given sets, then \(A \cap (A \cap B)^c\) is equal to (a) \(A\) (b) \(B\) (c) \(\Phi\) (d) \(A \cap B^c\) Solution \[ A\cap(A\cap B)^c \] Using De Morgan’s law, \[ =A\cap(A^c\cup B^c) \] \[ =(A\cap A^c)\cup(A\cap B^c) \] \[ =\Phi\cup(A\cap B^c) \] \[ =A\cap B^c \] Answer \[

If A and B are two sets such that \[ n(A)=70,\quad n(B)=60,\quad n(A\cup B)=110 \] Then, \(n(A\cap B)\) is equal to (a) 240 (b) 50 (c) 40 (d) 20 Solution Using the formula, \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \] Substituting the values, \[ 110=70+60-n(A\cap B) \] \[ 110=130-n(A\cap B) \] \[ n(A\cap B)=130-110 \] \[ =20

For two sets \(A \cup B = A\) iff (a) \(B\subseteq A\) (b) \(A\subseteq B\) (c) \(A\ne B\) (d) \(A=B\) Solution If \[ A\cup B=A \] then every element of \(B\) is already present in \(A\). Therefore, \[ B\subseteq A \] Answer \[ \boxed{B\subseteq A} \] Correct option: (a) Next Question / Full Exercise