Final answer:
To show that (A∩B)∪C=A∩(B∪C) if and only if C⊆A, we need to prove both directions of the statement.
Step-by-step explanation:
To show that (A∩B)∪C=A∩(B∪C) if and only if C⊆A, we need to prove both directions of the statement.
Forward direction:
If (A∩B)∪C=A∩(B∪C), then for any element x in C, x must also be in A∩(B∪C). This means that x is in A and either in B or C. Since x is in A, we can conclude that C⊆A.
Backward direction:
If C⊆A, then for any element x in C, x is also in A. This implies that x is in (A∩B)∪C. On the other hand, x is in A, and either in B or C, so x is in A∩(B∪C). Therefore, (A∩B)∪C=A∩(B∪C).