Question

For sets A, B and C by using properties of sets

(A union B) - C = (A – C) union (B - C)​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that,

$\rm :\longmapsto\:x \: \in \: (A\cup B) - C$

$\rm :\implies\:x \: \in \: \bigg((A\cup B)\cap {C}^{'} \bigg)$

$\rm :\implies\:x \: \in \: \bigg((A\cap {C}^{'})\cup (B\cap {C}^{'}) \bigg)$

$\rm :\implies\:x \: \in \: (A - C)\cup (B - C)$

$\rm :\implies\:(A\cup B) - C \: \sub \: (A - C)\cup (B - C) - - (1)$

Let assume that,

$\rm :\longmapsto\:x \: \in \: (A - C)\cup (B - C)$

$\rm :\implies\:x \: \in \: \bigg((A\cap {C}^{'})\cup (B\cap {C}^{'}) \bigg)$

[ Using Distributive Law ]

$\rm :\implies\:x \: \in \: \bigg((A\cup B)\cap {C}^{'} \bigg)$

$\rm :\implies\:x \: \in \: (A\cup B) - C$

$\rm :\implies\:(A - C)\cup (B - C) \: \sub \: (A\cup B) - C - - (2)$

From equation (1) and (2),

$\rm :\implies\:(A\cup B) - C \: = \: (A - C)\cup (B - C)$

Hence, Proved