Why does $|A|\le|B|$ imply $|\mathcal{P}(A)|\le|\mathcal{P}(B)|$? (Use elementary notations; for beginners to advanced math)

Question

Edit: @QuantamSpace’s answer contains advanced notation. I need notation that is more elementary, using Chapter 5.4 in A Transition To Advanced Math: 8th Edition

According to the answer key the following is true: If $|A|\le|B|$ then $2^{|A|}\le 2^{|B|}$.

For finite sets, if $|A|\le|B|$ then $2^{|A|}\le 2^{|B|}$. Since $|\mathcal{P}(A)|=2^{|A|}$ and $|\mathcal{P}(B)|=2^{|B|}$, $|\mathcal{P}(A)|\le|\mathcal{P}(B)|$. However, for infinite sets, does this hold?

Attempt (EDIT: My Attempt is wrong. You can ignore this part.)

According to my textbook I learned the following (using Cantor's Theorem):

$|A|\le \mathcal|{P}(A)|$ and $|B|\le \mathcal|{P}(B)|$.

Moreover, since $|A|\le|B|$, $f:A\xrightarrow{1-1}B$, $i:A\xrightarrow{1-1}\mathcal{P}(A)$ and $g:B\xrightarrow{1-1}\mathcal{P}(B)$.

To prove a counter example we need that $h:\mathcal{P}(B)\to\mathcal{P}(A)$ is one-to-one so that $|\mathcal{P}(B)|\le|\mathcal{P}(A)|$.

Now suppose if $|A|\le |B|$ then $|\mathcal{P}(A)|\le|\mathcal{P}(B)|$; also suppose $i=h \circ g \circ f$. We already know $i$ is one-to-one. Hence, since $h \circ g$ is one-to-one, $f$ is one-to-one. Therefore $\mathcal{P}(B)\le\mathcal{P}(A)$. This is a contradiction. Hence we have shown a counter example.

Question: Am I correct. Is the answer key wrong?

Answer 1

Note that the assumption $|A| \le |B|$ means that there is one-one function $f: A \to B$. Define a function $$F: \mathcal{P}(A) \to \mathcal{P}(B)$$ by $F(X)= f[X]:= \{f(x): x \in X\}$ for $X \in \mathcal{P}(A)$.

Then $F$ is one-one. To see this, assume that $X \ne Y$. Then without loss of generality, we may assume that $X \not\subseteq Y$ and thus there is $x \in X$ such that $x \notin Y$. Then note that $f[X] \ne f[Y]$, because if they would be equal, then $f(x) \in f[Y]$ so there would be $y \in Y$ with $f(x) = f(y)$ which would imply that $x=y\in Y$, since $f$ is injective. This contradicts the fact that $x \notin Y$.

That $F$ is injective means precisely that $|\mathcal{P}(A)| \le |\mathcal{P}(B)|$, and we are finished.

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Intuitively, you can expect this to be true as well: $|A| \le |B|$ means that you can embed $A$ as a subset of $B$, and a subset of $B$ is a subset of $A$ so we see that $\mathcal{P}(A)$ can be seen as a subset of $\mathcal{P}(B).$