show that (A ∪ B)\C ⊆ A ∪ (B\C)

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Let $x \in (A\cup B)\setminus C$ . Then,

$x \in A \cup B ; x \notin C \\ \Rightarrow x \in A ~\text{or}~ x \in B ; x \notin C\\ \Rightarrow x \in A\setminus C ~\text{or}~ x \in B\setminus C\\$

Since $x \in A~ \text{or~} x \in B \setminus C,$ we get $x \in A\cup (B \setminus C).$

Therefore,

$(A ∪ B)\setminus C ⊆ A ∪ (B\setminus C)$

show that (A ∪ B)\C ⊆ A ∪ (B\C)

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Question ID: mtid-5-stid-8-sqid-161-qpid-50