How to left align the beginnings of two equations and right align the ends of the equations

Define \ceilset and \ceilif according to your needs and try the following

\documentclass[paper=letter,twoside,fontsize=12pt,leqno]{scrbook}
\usepackage{mathtools}
\begin{document}
\noindent Define the following two functions:
\begin{align*}
&\wedge \colon \ceilset \times \ceilset \longrightarrow \ceilset & & \text{and} & \vee \colon \ceilset \times \ceilset \longrightarrow \ceilset &  \\
\intertext{for each $(A, B) \in \ceilset \times \ceilset $ by the assignments}
&\wedge \colon (A, B) \longmapsto A \cap B && \text{and} & \vee \colon (A, B) \longmapsto \ceilf{A \cup B} &\text{.}
\end{align*}
\end{document}

If you also want to align the operators:

\documentclass[paper=letter,twoside,fontsize=12pt,leqno]{scrbook}
\usepackage{mathtools}

\usepackage{showframe}
\let\ceilset=\relax
\let\ceilf=\relax
\begin{document}
Define the following two functions:
\begin{align*}
&\wedge \colon \ceilset \times \ceilset &&\longrightarrow \ceilset && \text{and} && \vee \colon \ceilset \times \ceilset &&\longrightarrow  \\
\intertext{for each $(A, B) \in \ceilset \times \ceilset$ by the assignments}
&\wedge \colon (A, B) &&\longmapsto A \cap B && \text{and} && \vee \colon (A, B) &&\longmapsto \ceilf{A \cup B} \quad\text{.} \\
\end{align*}
\end{document}

Use alignat, that makes pairs of “right aligned/left aligned” columns with no intervening space (that you can add yourself).

\documentclass{article}
\usepackage{amsmath}

\newcommand{\ceilset}{\mathsf{Ceil}}
\newcommand{\ceilf}[1]{\lceil #1 \rceil}

\begin{document}

Define the following two functions 
\begin{alignat*}{3}
&\wedge \colon \ceilset \times \ceilset \longrightarrow \ceilset
  &\qquad\text{and}\qquad &&
\vee \colon \ceilset \times \ceilset \longrightarrow \ceilset \\
\intertext{for each $(A, B) \in \ceilset \times \ceilset$ by the assignments}
&\wedge \colon (A, B) \longmapsto A \cap B
  &\qquad\text{and}\qquad &&
\vee \colon (A, B) \longmapsto \ceilf{A \cup B} \makebox[0pt][l]{.}
\end{alignat*}
These functions will be used shortly.

\end{document}

I set the period hanging right, so the symbols are aligned.

You should also considering aligning at the wedge and vee:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\ceilset}{\mathsf{Ceil}}
\newcommand{\ceilf}[1]{\lceil #1 \rceil}

\begin{document}

Define the following two functions
\begin{alignat*}{2}
&\wedge \colon \ceilset \times \ceilset \longrightarrow \ceilset
  &\qquad\text{and}\qquad &
\vee \colon \ceilset \times \ceilset \longrightarrow \ceilset \\
\intertext{for each $(A, B) \in \ceilset \times \ceilset$ by the assignments}
&\wedge \colon (A, B) \longmapsto A \cap B
  &\qquad\text{and}\qquad &
\vee \colon (A, B) \longmapsto \ceilf{A \cup B} .
\end{alignat*}
These functions will be used shortly.

\end{document}

Notes.

1. I only left the package amsmath that's essential for the code.
2. The colon after “functions” is wrong.