Answer to Question #300513 in Discrete Mathematics for ratna

Question

show that (A ∪ B)\C ⊆ A ∪ (B\C)

Answer 1

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,

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