We want to determine whether or
not for all A, B, C ⊆ U,
A ∩ (B ∩ C)' = (A - B) U (A -
C) is an identity.
Which one of the following
alternatives is the correct way to
determine this?
Select one:
a.
x ∈ A ∩ (B ∩ C)'
iff x ∈ A and x ∈ (B ∩ C)'
iff x ∈ A and x ∈ (B ∩ C)
iff x ∈ A and (x ∉ B or x ∉ C)
iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C)
iff x ∈ (A - B) or x ∈ (A - C)
iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation.
We can therefore say that the given statement is an identity.
b.
x ∈ A ∩ (B ∩ C)'
iff x ∈ A and x ∈ (B ∩ C)'
iff x ∈ A and x ∈ (B - C)
iff x ∈ A and (x ∉ B or x ∉ C)
iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C)
iff x ∈ (A - B) or x ∈ (A - C)
iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation.
We can therefore say that the given statement is an identity.