# Thread: Image and Inverse Images for Projection Functions

1. ## Image and Inverse Images for Projection Functions

Let A and B be nonempty sets and $\pi_1 : A \times B \rightarrow A$ be the projection function $\pi_1 (x,y) = x$ for $(x,y) \epsilon A \times B$.

(a) For $C \subset A$, determine the inverse image set $\pi^{-1} (C)$.

(b) For $C \subset A, D \subset B$ and $D \neq \emptyset$, determine the image $\pi_1 (C \times D)$.

2. Originally Posted by onemore
Let A and B be nonempty sets and $\pi_1 : A \times B \rightarrow A$ be the projection function $\pi_1 (x,y) = x$ for $(x,y) \epsilon A \times B$.

(a) For $C \subset A$, determine the inverse image set $\pi^{-1} (C)$.

(b) For $C \subset A, D \subset B$ and $D \neq \emptyset$, determine the image $\pi_1 (C \times D)$.
You probably mean $\pi_1^{-1} (C)$, not $\pi^{-1} (C)$. Have you at least tried to solve this problem?

I thinkt the if you understand the definitions and if you draw a picture or think for a while, you see immediately that:
$\pi_1^{-1} (C) = C\times B$and
$\pi_1 (C\times D) = C$.

One question, to see whether you understand the problem - can you explaine why $D \ne \emptyset$ is needed in (b)?