# Proof theorem direct proof

For each theorem below, state whether or not the theorem is true and give either a direct proof, proof by cases or counter example to support your view
Theorem: For any sets $A$ and $B$ such that $A\subseteq B$ we have $\mathcal P(A)\subseteq \mathcal P(B)$

Here's my attempt:

$A=\{2,5,4\}\ B=\{1,2,4,5\}$ therefore $A⊆B$

$\mathcal P(B)=\{\{\},\{1\},\{2\},\{3\},\{4\},\{5\},\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{2,2\},\{2,3\},\{2,4\},\{2,5\},\{3,3\},\{3,4\},\{3,5\},\{4,4\},\{4,5\},\{5,5\}$ $\mathcal P(A)=\{\{\},\{2\},\{5\},\{4\},\{2,5\},\{2,4\},\{4,5\}\}$

So from my test I got direct proof

1. By assumption $A⊆$ know all elements of A are in set B
2. $\mathcal P(A)$ to be all subsets of $A$ and $\mathcal P(B)$ to be all subsets of $B$
3. by $\mathcal P(A) = \{1,2,5\}$ and $\mathcal P(B) = \{1,2,3,4,5\}$

All $A$ are in $B = \{\{\},\{1\},\{2\},\{3\},\{4\},\{2,5\},\{2,4\},\{4,5\}\}$ are all included in $\mathcal P(B)$

also it's homework, so please don't provide straight answers; just indication : )

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Please write in Latex and put your questions in text on stack exchange. Using an external link, and even an image is completely unnecessary in this case. – Carry on Smiling Dec 5 '13 at 22:06
You gave an example. An example is not a proof. A proof must show the truth of the statement in all cases — here, that would be for all sets $A, B$ such that $A \subseteq B$ — not just one case you selected. – Daniel Hast Dec 5 '13 at 22:06

Hint: $X \in \mathcal{P}(A)$ means by definition that $X \subset A$.

If $X \in \mathcal{P}(A)$, then $X \subset A \subset B$, thus $X \subset B$, then $X \in \mathcal{P}(B)$. Therefore $\mathcal{P}(A) \subset \mathcal{P}(B)$.

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$A\subseteq B \iff$for any $x\in A\rightarrow x\in B$

let $W$ be an element of the power set of $A$. Then that means for any $a\in W \rightarrow a\in A$.

is $a$ also in $B$??

would that mean $W$ is also in the power set of $B$?

would that mean $\mathcal{P}(A) \subseteq \mathcal{P}(B)$???

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woops, had some typos, sorry about that – Carry on Smiling Dec 5 '13 at 22:14

Hint Use the definition of $\subseteq$ and the fact that this is a transitive relationship.

Use:

1. $A\subseteq B \iff \forall x, x\in A\Rightarrow x\in B$,
2. $A\subseteq B\text{ and }B\subseteq C\iff A\subseteq C$,
3. $A\in\mathscr P(B)\iff A\subseteq B$.
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