Union Distributes over Intersection

Theorem
Set union is distributive over set intersection:


 * $$R \cup \left({S \cap T}\right) = \left({R \cup S}\right) \cap \left({R \cup T}\right)$$

Generalized Result

 * $$\forall n \in \N^*: S \cup \bigcap_{i = 1}^n T_i = \bigcap_{i = 1}^n \left({S \cup T_i}\right)$$

Proof
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Generalized Proof
For all $$n \in \N^*$$, let $$P \left({n}\right)$$ be the proposition: $$S \cup \bigcap_{i = 1}^n T_i = \bigcap_{i = 1}^n \left({S \cup T_i}\right)$$.


 * $$P(1)$$ is true, as this just says $$S \cup T_1 = S \cup T_1$$.


 * $$P(2)$$ is the case $$S \cup \left({T_1 \cap T_2}\right) = \left({S \cup T_1}\right) \cap \left({S \cup T_2}\right)$$ which has been proved. This is our basis for the induction.


 * Now we need to show that, if $$P \left({k}\right)$$ is true, where $$k \ge 2$$, then it logically follows that $$P \left({k+1}\right)$$ is true.

So this is our induction hypothesis:

$$S \cup \bigcap_{i = 1}^k T_i = \bigcap_{i = 1}^k \left({S \cup T_i}\right)$$

Then we need to show:

$$S \cup \bigcap_{i = 1}^{k+1} T_i = \bigcap_{i = 1}^{k+1} \left({S \cup T_i}\right)$$

This is our induction step:

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So $$P \left({k}\right) \implies P \left({k+1}\right)$$ and the result follows by the Principle of Mathematical Induction.