Final answer:
The student’s question about set operations has been verified in two parts, showing that both A - (B U C) = (A - B) ∩ (A - C) and A - (B ∩ C) = (A - B) U (A - C) hold true by determining the sets involved through step-by-step set operation calculations.
Step-by-step explanation:
To verify the given statements, we’ll perform the set operations step by step. Firstly, the union and intersection operations, and then the set difference operations as per the questions.
For (i) A - (B U C):
B U C = {1, 2, 3, 4, 5, 6, 7, 10} (the elements that are in B or C or in both). So A - (B U C) is the set of elements that are in A but not in (B U C), which equals {8}.
For (A-B):
A-B = {2, 4, 6, 8, 10} - {1, 2, 3, 4, 5, 6, 7} which equals {8, 10}. Then (A - B) ∩ (A - C) = {8, 10} ∩ {2, 4, 8} = {8} which is the same as A - (B U C).
Therefore (i) is verified as true.
For (ii) A - (B ∩ C):
B ∩ C = {2, 6, 7} (elements that are in both B and C). So A - (B ∩ C) is the set of elements that are in A but not in (B ∩ C), which equals {4, 8, 10}.
For (A - B):
We already found (A - B) = {8, 10} and we previously have A - C = {2, 4, 8}. Taking the union, (A - B) U (A - C) = {2, 4, 8, 10}, which is the same as A - (B ∩ C).
Therefore (ii) is verified as true.