A\B ⊆ C ∩ D and x ∈ A . I must show that if x $\notin$ D then x $\in$ B. So far, I am thinking that, $$\text{x} \in \text{A }\land \text{x} \notin \text{B} \implies \text{x}\in C \land \text{x}\in D $$ So if $$\text{x} \notin \text{D} \implies \text{x}\notin A $$, which means it is in B. I was wondering if this sort of logical thinking was valid or not.
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Yes it is; your $x$ is in $A$, and you wrote that if $x$ were also not in $B$ then it would be in $D$; so since your $x$ is not in $D$, necessarily it must lie in $B$. – 18cyclotomic Jun 17 '17 at 20:00
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@18cyclotomic ok thank you so much – Math2Hard Jun 17 '17 at 20:04
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Your logic is good, although it would be better if you'd underline the fact that $x \notin D$ implies $x \notin C \cap D$. – theSongbird Jun 17 '17 at 20:43