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Homework Statement
I'm having issues understanding a mistake that I'm making, any assistance is appreciated! I know a counterexample but my attempt at proving the proposition is what's troubling me.
Prove or disprove
$$P(A \cup B) \subseteq P(A) \cup P(B) $$
Homework Equations
The Attempt at a Solution
Let ##C## be an particularly, arbitrarily picked element of element of ##P(A \cup B)##
Then by definition of a powerset
$$C \subseteq A \cup B$$
This means that
$$\forall x \in C, \ x \in A \cup B$$
$$\forall x \in C, \ x \in A \ \text{or} \ x \in B \ \ \ \ ^{**} $$
$$\forall x \in C, \ x \in A \ \text{or} \ \ \ \forall x \in C , x \in B \ \ \ \ ^{**}$$
Then ##C \subseteq A## or ##C \subseteq B ## and thus ##C \in P(A) \ \text{or} \ C \in P(B)## and ##C \subseteq P(A) \cup P(B)##. The bits I left the asterisks were where I felt I made some error.
But, in a proof of a similar problem: ##P(A \cap B) \subseteq P(A) \cap P(B) ##, I wrote
$$\forall x \in C, \ x \in A \cap B$$
$$\forall x \in C, \ x \in A \ \text{and} \ \forall x \in C, x \in B $$
Then ##C \subseteq A## and ##C \subseteq B ##, and thus ##C \in P(A) \ \text{and} \ C \in P(B)##, and ##C \subseteq P(A) \cap P(B)##. This gave me the right answer.
Why did the first one give me the wrong answer while the second one gave a right one?
PS It would be nice if the explanation isn't too technical as it's my first course on the subject. Thanks!