I was just looking over my notes and found that some steps, while obvious, seem to be missing in-between steps. For example:
$$x ∈ S \land (x ∈ T \lor x ∈ V) \Rightarrow (x ∈ S \land x ∈ T) \lor (x ∈ S \land x ∈ V)$$
Now this step seems intuitive. I mean, if $x$ belongs to $S$ and $x$ either belongs to $T$ or $V$, then obviously either $x$ belongs to $S$ and $T$, or $x$ belongs to $S$ and $V$.
But given that the previous steps were just converting from $x ∈ S ∩ (T ∪ V)$ to $x ∈ S \land (x ∈ T \lor x ∈ V)$, it seems like a lot more is being done in this step than simply following definitions of union and intersection. Therefore, is this step missing an intermediate step? Or is there a logical principle that I'm not aware of that allows this to happen in one step?
$\endgroup$1 Answer
$\begingroup$It is the distributive property of unions and intersections. Or equivalently, that of "and" and "or."
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