Let A, B and C denotes the subset of a set, S and let 𝐶̅denotes the complement of C in set S. If (A ∩ C) = (B ∩ C) and (A ∩ 𝐶̅) = (B ∩ 𝐶̅), then prove that (A = B).

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Since (A ∩ C) = (B ∩ C) and (A ∩ 𝐶̅) = (B ∩ 𝐶̅) then,

(B ∩ C) - (A ∩ C)= $\empty$ and (A ∩ 𝐶̅) - (B ∩ 𝐶̅)= $\empty$

Therefore, (B ∩ C) - (A ∩ C) = (A ∩ 𝐶̅) - (B ∩ 𝐶̅)

This can also be written as,

(A ∩ C) + (A ∩ 𝐶̅)= (B ∩ C)+ (B ∩ 𝐶̅)

This can be represented using union as follows,

$(A\cap C)\cup(A\cap C')=(B\cap C)\cup(B\cap C')$

Recognize that the left hand side is A since A = $(A\cap C)\cup(A\cap C')$ and the right hand side is B since B = $(B\cap C)\cup(B\cap C').$

This shows that, A=B as required.