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Appendix A Notation
| Symbol |
Description |
Location |
| \(x\in S\) |
\(x\) is an element of \(S\)
|
Paragraph |
| \(x\notin S\) |
\(x\) is not an element of \(S\)
|
Paragraph |
| \(\{x\in S : P(x)\}\) |
\(x\) in \(S\) such that \(x\) has property \(P\)
|
Paragraph |
| \(A\subseteq S\) |
\(A\) is a subset of \(S\)
|
Definition 1.2.2 |
| \(A\nsubseteq S\) |
\(A\) is not a subset of \(S\)
|
Paragraph |
| \(\mathbb{R}\) |
the set of real numbers |
Item |
| \(\mathbb{Z}\) |
the set of integers |
Item |
| \(\mathbb{Q}\) |
the set of rational numbers |
Item |
| \(\mathbb{N}\) |
the set of natural numbers |
Item |
| \(\mathbb{Z}^+\) |
the set of positive integers |
Item |
| \(\mathbb{Z}^{nonneg}\) |
the set of nonnegative integers |
Item |
| \(\mathbb{R}^+\) |
the set of positive real numbers |
Item |
| \(\mathbb{R}^{nonneg}\) |
the set of nonnegative real numbers |
Item |
| \(A\times B\) |
the product of \(A\) and \(B\text{;}\) \(\{(a, b) : a\in A, b\in B\}\)
|
Definition 1.2.6 |
| \(aRb\) |
\(a\) is related to \(b\)
|
Paragraph |
| \(\sim p\) |
not \(p\)
|
Item |
| \(p\wedge q\) |
\(p\) and \(q\)
|
Item |
| \(p\vee q\) |
\(p\) or \(q\)
|
Item |
| \(\mathbf{t}\) |
a statement that is always true; tautology |
Paragraph |
| \(\mathbf{c}\) |
a statement that is always false; contradiction |
Paragraph |
| \(P\equiv Q\) |
\(P\) is logically equivalent to \(Q\)
|
Definition 2.1.10 |
| \(p\rightarrow q\) |
if \(p\) then \(q\)
|
Item |
| \(\therefore\) |
therefore |
Assemblage |
| \(\forall\) |
for all; universal quantifier |
Item |
| \(\exists\) |
there exists; existential quantifier |
Item |
| \(\mathbb{Q}\) |
the set of ratioanl numbers |
Definition 4.2.1 |
| \(\mathbb{R}\setminus\mathbb{Q}\) |
the set of irratioanl numbers |
Definition 4.2.2 |
| \(d\mid n\) |
\(d\) divides \(n\)
|
Paragraph |
|
\(d\) does not divide \(n\)
|
Paragraph |
| \(n \text{ div } d\) |
quotient when \(n\) is divided by \(d\)
|
Paragraph |
| \(n \text{ mod } d\) |
remainder when \(n\) is divided by \(d\)
|
Paragraph |
| \(\sum_{k=1}^{n}a_k\) |
the sum of \(a_k\) from \(k=1\) to \(n\)
|
Assemblage |
| \(\prod_{k=1}^{n}a_k\) |
|
Paragraph |
| \(\binom{n}{r}\) |
\(n\) choose \(r\)
|
Definition 5.1.6 |
| \(A\cup B\) |
\(A\) union \(B\)
|
Definition 6.1.4 |
| \(A\cap B\) |
\(A\) intersect \(B\)
|
Definition 6.1.6 |
| \(A-B\) |
\(A\) minus \(B\text{;}\) the difference of set \(A\) and \(B\)
|
Definition 6.1.8 |
| \(A^C\) |
the complement of \(A\)
|
Definition 6.1.10 |
| \(\bigcup_{i=1}^{n}A_i\) |
the union \(A_1\cup A_2\cup\cdots \cup A_n\)
|
Paragraph |
| \(\bigcap_{i=1}^{n}A_i\) |
the intersection \(A_1\cap A_2\cap\cdots \cap A_n\)
|
Paragraph |
| \(\mathcal{P}(A)\) |
the power set of \(A\)
|
Definition 6.1.13 |
| \(\Leftrigharrow\) |
if and only if in proofs |
Paragraph |
| \(|S|\) |
the number of elements in \(S\)
|
Paragraph |
| \(\text{Im}(f)\) |
the image of \(f\)
|
Definition 7.1.6 |
| \(f^{-1}(x)\) |
the inverse of function \(f\)
|
Theorem 7.2.13 |
| \(x R y\) |
\(x\) is related to \(y\)
|
Paragraph |
| \(m\equiv n \mod d\) |
\(m\) is congruent to \(n\) mod \(d\text{;}\) \(d\mid (m-n)\)
|
Paragraph |
| \([a]\) |
the equivalence class of \(a\)
|
Paragraph |
| \(P(n, r)\) |
the number of \(r\)-permutations from a set of \(n\) elements |
Definition 9.2.8 |
| \(\binom{n}{r}\) |
\(n\) choose \(r\)
|
Definition 9.5.1 |
