Answer to Question #252170 in Discrete Mathematics for Wafa Abbas

Question

Use algebra of sets to prove that,

 [(𝐵 − 𝐴)' ∩ 𝐴] − 𝐴' = 𝐴

Answer 1

Solution:

$LHS=[(𝐵 − 𝐴)' ∩ 𝐴] − 𝐴' \\=[(𝐵' − 𝐴')∩ 𝐴] − 𝐴'$

$=[(𝐵' − 𝐴') ∩ 𝐴] ∩ [𝐴']' \ \ [\because P-Q=P∩Q']$

$=[(𝐵' − 𝐴') ∩ 𝐴] ∩ A \\=(𝐵' − 𝐴') ∩ 𝐴∩ 𝐴 \\=(𝐵' − 𝐴') ∩ 𝐴$

$\\=(𝐵 − 𝐴)' ∩ 𝐴 \\=A-(B-A) \\=A \\=RHS$

Hence, proved.