Please show all the steps to solve this problem For any inte

Please show all the steps to solve this problem.

For any integer n lessthanorequalto 1, if A_1, A_2, A_3, ..., A_n and B are any sets, then (A_1 - B) (A_2 - B) ... (A_n - B) = (A_1 A_2 A_3 ... A_n) - B.

Solution

Let us prove by induction on n.

For n=1, trivially true since A₁- B= A₁-B

induction hypothesis: let it be true for n=k-1

i.e. (A₁-B) U (A₂-B) U... U (A_k-1-B)= (A₁UA₂U... UA_k-1) - B

To prove for n=k

Consider (A₁-B) U (A₂-B) U... U (A_k-1-B) U (A_k-B)

=  ((A₁UA₂U... UA_k-1) - B ) U (A_k-B) ...[from induction hypothesis]

= (A₁UA₂U... UA_k-1UA_k) - B ...[since for all sets A,B & C, (A-B) U (C-B) = (A U C) - B;

here, taking A₁UA₂U... UA_k-1 = A & A_k= C (union of sets is a set)]

Thus, we have proved the statement is true for all positive integer n.

hence, (A₁-B) U (A₂-B) U...   U (An-B) =   (A₁UA₂U... UAn) - B