Power Sets

**GoldendoodleMom** · September 22nd 2008, 07:47 AM

Prove: Let S and T be sets. If there exists and injection from |S| --> |T| then there exists an injection from |P(S)|-->|P(T)|

(P(S) and P(T) represent the power sets of each)

**Plato** · September 22nd 2008, 08:15 AM

Here is a bit of notation. $C \subseteq S \Rightarrow \quad \overrightarrow f (C) = \left\{ {t \in T:\left( {\exists x \in C} \right)\left[ {f(x) = t} \right]} \right\}$
That is called the image set of $C$ under $f$ .
Now it is clear that we have $\overrightarrow f :P(S) \to P(T)$ , so we must prove that if $f\mbox { is injective then } \overrightarrow f \mbox{ is also injective}$
Of course to do that you must show $\overrightarrow f (C) = \overrightarrow f (D) \Rightarrow \quad C = D$ .

**GoldendoodleMom** · September 22nd 2008, 08:35 AM

The proof is regarding the cardinality of the sets and power sets. Does that change the strategy?

**Plato** · September 22nd 2008, 08:43 AM

Originally Posted by GoldendoodleMom

The proof is regarding the cardinality of the sets and power sets. Does that change the strategy?

No it does not.
Here is the rest of it.
If $\overrightarrow f (C) = \overrightarrow f (D)$ then
$\begin{array}{lcl} {c \in C} & \Rightarrow & {f(c) \in \overrightarrow f (C)} \\ {} & \Rightarrow & {f(c) \in \overrightarrow f (D)} \\ {} & \Rightarrow & {\left( {\exists d \in D} \right)\left[ {f(d) = f(c)} \right]} \\ {} & \Rightarrow & {d = c} \\ {} & \Rightarrow & {c \in D} \\ \end{array}$
You do the other way by switching D with C.

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