Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C:
Explanation
A universal set must contain all elements of the given sets A, B, and C as subsets. The combined elements of A = {1, 3, 5}, B = {2, 4, 6}, and C = {0, 2, 4, 6, 8} are {0, 1, 2, 3, 4, 5, 6, 8}. Option (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} contains all these elements, making it a valid universal set. Option (i) lacks element 8 from set C. Option (ii) φ is empty and cannot contain any non-empty sets. Option (iv) lacks element 0 from set C.
Solution Steps
Step 1: Identify all elements in sets A, B, C: A = {1, 3, 5}, B = {2, 4, 6}, C = {0, 2, 4, 6, 8}
Step 2: Combined elements needed: {0, 1, 2, 3, 4, 5, 6, 8}
Step 3: Check each option to see if it contains all required elements
Step 4: Option (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} contains all elements of A, B, and C, so it qualifies as universal set