I am using an align environment to display two expressions on each of two lines. I would like the first expression in each line to be left aligned and the second expression in each line to be right aligned. The first expression in each line are aligned but I can't get the second expression in each line right aligned. The quads are used to reduce the empty space in the center of each line.
\documentclass[paper=letter,twoside,fontsize=12pt,leqno]{scrbook}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{microtype}
\usepackage{enumitem}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{scrlayer-scrpage}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{mathtools}
\newcommand{\ceilset}{\mathsf{Ceil}}
\newcommand{\ceilf}[1]{\lceil #1 \rceil}
\begin{document}
Define the following two functions:
\begin{align*}
\quad \qquad \qquad &\wedge \colon \ceilset \times \ceilset \longrightarrow \ceilset & \text{and} && \vee \colon \ceilset \times \ceilset &\longrightarrow \ceilset \qquad \qquad \quad \\
\intertext{for each $(A, B) \in \ceilset \times \ceilset$ by the assignments}
\quad \qquad \qquad &\wedge \colon (A, B) \longmapsto A \cap B & \text{and} && \vee \colon (A, B) &\longmapsto \ceilf{A \cup B} \text{.} \qquad \qquad \quad
\end{align*}
These functions will be used shortly.
\end{document}
If my proposed alignment is a bad idea, please suggest an alternative.
\ceilset
and\ceilf
commands?