Master Operations on Sets for CBSE Class 11 Applied Mathematics. Learn Union, Intersection, Difference, and Complements with solved examples and exercises.
The union of two sets A and B, denoted by A ∪ B, is the set of all elements which are in A, in B, or in both. The intersection, denoted by A ∩ B, consists of elements common to both sets. These operations are fundamental for analyzing overlapping data groups in applied mathematics.
Step 1: Given A = {1, 2, 3} and B = {3, 4, 5}.
Step 2: A ∪ B = {1, 2, 3, 4, 5}.
Step 3: A ∩ B = {3}.
Answer: A ∪ B = {1, 2, 3, 4, 5}, A ∩ B = {3}
The difference of sets A and B, denoted by A − B, contains elements that are in A but not in B. The complement of a set A, denoted by Aᶜ or A', contains all elements of the universal set U that are not in A. These operations help isolate specific subsets within a defined universe.
Step 1: Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}.
Step 2: Aᶜ = U − A = {4, 5, 6}.
Step 3: If B = {2, 3, 4}, then A − B = {1}.
Answer: Aᶜ = {4, 5, 6}, A − B = {1}
When dealing with three sets A, B, and C, the Principle of Inclusion-Exclusion allows us to find the total number of elements in their union. This is essential for solving complex survey problems involving multiple categories. It accounts for double and triple counting of overlapping elements.
Step 1: Given n(A)=10, n(B)=12, n(C)=15, n(A∩B)=5, n(B∩C)=4, n(A∩C)=3, n(A∩B∩C)=2.
Step 2: Sum = 10 + 12 + 15 = 37.
Step 3: Subtract intersections = 37 − (5 + 4 + 3) + 2 = 37 − 12 + 2 = 27.
Answer: 27