# Not Just Calculus: First-Year Introduction to Upper-Level Mathematics

## ReferencesNotation

Symbol Description Location
$$x\in A$$ $$x$$ is an element of $$A$$ Paragraph
$$\not\in$$ $$x$$ is not an element of $$A$$ Paragraph
$$\{\ldots\}$$ curly brackets Paragraph
$$\emptyset$$ the empty set Paragraph
$$A \subseteq B$$ $$A$$ is a subset of $$B$$ Paragraph
$$A \not\subseteq B$$ $$A$$ is not a subset of $$B$$ Paragraph
$$A \cap B$$ intersection of sets $$A,B$$ Paragraph
$$A \cup B$$ union of sets $$A,B$$ Paragraph
$$A \setminus B$$ difference of sets $$A,B$$ Paragraph
$$(x,y)$$ ordered pair Paragraph
$$A\times B$$ the (Cartesian) product of sets $$A,B$$ Paragraph
$$f\colon S\to T$$ a function $$f$$ with domain $$S$$ and codomain $$T$$ Paragraph
$$f(s)=t$$ $$(s,t)$$ is an element of $$f$$ Paragraph
$$g\of f$$ composition of functions $$g,f$$ Paragraph
$$I_S$$ identity function on a set $$S$$ Paragraph
$$\Id_S$$ identity function on a set $$S$$ Paragraph
$$\One_S$$ identity function on a set $$S$$ Paragraph
$$f(U)$$ the image of the set $$U$$ under the function $$f$$ Paragraph
$$f^{-1}(V)$$ the preimage of the set $$V$$ under the function $$f$$ Paragraph
$$|S|$$ size of a finite set $$S$$ Paragraph
$$\Z$$ the set of integers Paragraph
$$a|b$$ $$a$$ divides $$b$$ Paragraph
$$a\nmid b$$ $$a$$ does not divide $$b$$ Paragraph
$$a\equiv b \pmod{m}$$ $$a$$ is equivalent to $$b$$ modulo $$m$$ Paragraph
$$\Z_m$$ the set $$\{0,1,\ldots,m-1\}$$ Paragraph
$$n \MOD m$$ $$n$$ modulo $$m$$ Paragraph
$$\Q$$ the set of rational numbers Paragraph
$${\mathcal P}(A)$$ power set of a set $$A$$ Exercise 1.5.3.1
$$\chi_A$$ characteristic function for the set $$A$$ Exercise 1.5.3.3
$${\mathbf e}_1,{\mathbf e}_2$$ standard basis vectors in $$\R^2$$ Paragraph
$$\left\Vert(a,b)\right\Vert$$ norm of the ordered pair $$(a,b)$$ Paragraph
$$S^1$$ unit circle Paragraph
$$[L]$$ matrix for a linear map $$L$$ Paragraph
$$\det$$ determinant of a matrix Paragraph
$$\re(z)$$ real part of the complex number $$z$$ Paragraph
$$\im(z)$$ imaginary part of the complex number $$z$$ Paragraph
$$|z|$$ norm of the complex number $$z$$ Paragraph
$$\arg(z)$$ argument of the complex number $$z$$ Paragraph
$$i$$ the complex number $$i$$ Paragraph
$$\overline{z}$$ conjugate of the complex number $$z$$ Paragraph
$${z}^\ast$$ conjugate of the complex number $$z$$ Paragraph
$$\proj_{\mathbf{u}}\mathbf{v}$$ orthogonal projection of vector $$\mathbf{v}$$ onto vector $$\mathbf{u}$$ Exercise 2.4.1.2