Set theory practice · Medium

Option B (B) is correct because if $A \cup B = A$, it implies that B is a subset of A, and therefore, the intersection of A and B is B.

Option A ($B \cap C^{\prime}$) is correct because the given set expression simplifies to $B \cap C^{\prime}$ using set identities.

Option A (300) is correct because it is the number of integers in the given range that are neither multiples of 3 nor multiples of 4.

Option C ($\frac{1}{\log _{\mathrm{xy}} \mathrm{xyz}}+\frac{1}{\log _{\mathrm{yz}} \mathrm{xyz}}+\frac{1}{\log _{\mathrm{zx}} \mathrm{xyz}}=2$) is correct because using the change of base formula and properties of logarithms, the expression simplifies to 2.

Option D (3) is correct because it represents the number of elements in the complement of the union of sets A and B, calculated by subtracting the number of elements in the union from the total number of elements in the universal set.

Option B (15) is correct because it is the number of people who invest in both National Savings Certificates and shares, calculated using set theory principles.

Option B (8) is correct because the number of subsets containing 2, 3, and 5 is the same as the number of subsets of the remaining elements, which is 2^(6-3) = 2³ = 8.

Option A is correct because the minimum number of elements in A ∪ B occurs when A is a subset of B, resulting in A ∪ B having the same number of elements as B, which is 6.

Option D (U) is correct because A contains all even numbers, B contains all odd numbers, their intersection is empty, and the complement of an empty set is the universal set U.

Option B ($A-(B \cup C)=(A \cap B^{\prime}) \cap C^{\prime}$) is correct because it represents the set of elements in A that are not in B and not in C, which is equivalent to the intersection of A with the complements of B and C.

Option A ({ (1,3),(2,3),(3,1),(3,2),(1,1),(2,1),(1,2) }) is correct because it accurately represents the union of the Cartesian products A × B and B × A.

Option C (Both I and II are true) is correct because the complement of a subset of the universal set is also a subset, and the double complement of a set is the set itself.

Option A (The set of isosceles triangles; the set of non-isosceles right angled triangles) is correct because the intersection of isosceles and equilateral triangles is the set of isosceles triangles, and right-angled triangles excluding isosceles triangles are non-isosceles right-angled triangles.

Option D (6) is correct because by finding all possible integer pairs (a, b) satisfying the equation and then filtering those where a > b, we arrive at a total of 6 elements in the intersection.

Option B (3300) is correct because it represents the number of families that buy newspaper A only, calculated using the principle of inclusion-exclusion.

Option A ($\\{x:|x|=5, x \in N\\}$) is correct because it results in a set with only one element, x = 5, satisfying the condition.

Option B (8) is correct because it represents the number of teachers teaching only Physics, calculated by subtracting the number of teachers teaching both subjects from the total number of Physics or Mathematics teachers.

Option B ({1,9},{1,9},{3,4,5,6,7,8}) is correct because it accurately calculates the complement of the union of A and B, the intersection of the complements of A and B, and the symmetric difference of A and B.