Decide, among the following sets, which sets are subsets of one and another: A = {x : x ∈ R and x satisfy x² – 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, ...}, D = {6}.
First, we find the elements of each set:
Set A: Solving x² – 8x + 12 = 0, we get (x – 6)(x – 2) = 0, so x = 2, 6. Thus, A = {2, 6}.
Set B = {2, 4, 6}
Set C = {2, 4, 6, 8, ...} (even natural numbers)
Set D = {6}
Now checking subset relations:
- D ⊂ A, since {6} ⊂ {2, 6}
- D ⊂ B, since {6} ⊂ {2, 4, 6}
- D ⊂ C, since {6} ⊂ {2, 4, 6, 8, ...}
- A ⊂ B, since {2, 6} ⊂ {2, 4, 6}
- A ⊂ C, since {2, 6} ⊂ {2, 4, 6, 8, ...}
- B ⊂ C, since {2, 4, 6} ⊂ {2, 4, 6, 8, ...}
Therefore, D ⊂ A ⊂ B ⊂ C and D ⊂ B ⊂ C.
Explanation
This question tests understanding of subsets. First, students must solve the quadratic equation to find elements of Set A. Then they must compare each set with others to identify subset relationships. The key insight is that D is a subset of A, B, and C; A is a subset of B and C; and B is a subset of C.
Solution Steps
Step 1: Solve x² – 8x + 12 = 0 to find A = {2, 6}
Step 2: List all sets: A = {2, 6}, B = {2, 4, 6}, C = {2, 4, 6, 8, ...}, D = {6}
Step 3: Check if each element of one set belongs to another set
Step 4: Identify all subset relationships: D ⊂ A ⊂ B ⊂ C