Let's start by defining the sets involved in the problem:
- [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]
- [tex]\( B = \{2, 3\} \)[/tex]
- [tex]\( C = \{3, 4\} \)[/tex]
We are to verify that [tex]\( A - (B \cup C) = (A - B) \cap (A - C) \)[/tex].
Step 1: Calculate [tex]\( B \cup C \)[/tex]
The union of sets [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ B \cup C = \{2, 3\} \cup \{3, 4\} = \{2, 3, 4\} \][/tex]
Step 2: Calculate [tex]\( A - (B \cup C) \)[/tex]
The difference between set [tex]\( A \)[/tex] and the union [tex]\( B \cup C \)[/tex]:
[tex]\[ A - (B \cup C) = \{1, 2, 3, 4, 5\} - \{2, 3, 4\} = \{1, 5\} \][/tex]
Step 3: Calculate [tex]\( A - B \)[/tex]
The difference between set [tex]\( A \)[/tex] and set [tex]\( B \)[/tex]:
[tex]\[ A - B = \{1, 2, 3, 4, 5\} - \{2, 3\} = \{1, 4, 5\} \][/tex]
Step 4: Calculate [tex]\( A - C \)[/tex]
The difference between set [tex]\( A \)[/tex] and set [tex]\( C \)[/tex]:
[tex]\[ A - C = \{1, 2, 3, 4, 5\} - \{3, 4\} = \{1, 2, 5\} \][/tex]
Step 5: Calculate [tex]\( (A - B) \cap (A - C) \)[/tex]
The intersection of the differences calculated in steps 3 and 4:
[tex]\[ (A - B) \cap (A - C) = \{1, 4, 5\} \cap \{1, 2, 5\} = \{1, 5\} \][/tex]
Conclusion:
We have:
[tex]\[ A - (B \cup C) = \{1, 5\} \][/tex]
and
[tex]\[ (A - B) \cap (A - C) = \{1, 5\} \][/tex]
Both sides of the equation are equal, hence:
[tex]\[ A - (B \cup C) = (A - B) \cap (A - C) \][/tex]
Thus, we have verified that the given statement is true.
