# Thread: Proving a certain function is injective

1. ## Proving a certain function is injective

Let $A$ and $B$ be nonempty sets such that $|A| < |B|$. Show there exists an injective function from $\mathcal{P}(A)$ to $\mathcal{P}(B)$.

This was a question on my final and luckily I was able to omit it because I wasn't sure how to tackle this problem, but I'm curious to what the answer is. Thanks!

2. Originally Posted by Pinkk
Let $A$ and $B$ be nonempty sets such that $|A| < |B|$. Show there exists an injective function from $\mathcal{P}(A)$ to $\mathcal{P}(B)$.

This was a question on my final and luckily I was able to omit it because I wasn't sure how to tackle this problem, but I'm curious to what the answer is. Thanks!
so there exists an injection $f: A \to B.$ noe define $g :\mathcal{P}(A) \to \mathcal{P}(B)$ by $g(X)=f(X)=\{f(x): \ x \in X \},$ for all $X \in \mathcal{P}(A).$