Proving whether a given function involving power sets is injective or surjective

Question

Let A = {1,2} and B = {2,3}.

Consider $\varphi : \wp(A \cup B) \to \wp(A)$ defined by

$$\varphi: U \mapsto U \cap A$$

for every $U \subseteq A \cup B$.

For example, $\varphi(\emptyset)= \emptyset$ and $\varphi$({1,3}) = {1}.

What, though, would the value of $\varphi$({2,3}) be? Is $\varphi$ injective or surjective? Me needs help proving this final part.

I know normally you would try to find that x = y to prove injectivity but I'm not even sure this function actually is injective.

Answer 1

First question: $$\phi(\{2,3\})=\{2,3\}\cap\{1,2\}=\cdots?$$

Hints for the rest.

• For injectivity: how many elements are there in the domain of $\phi$? How many in the codomain? Is it possible that all inputs give different outputs?
• For surjectivity: what is $\phi(U)$ when $U$ is a subset of $A$?

See if you can finish this.