## Notation

This table defines the notation used in this course.

 Symbol Description x ∈ A x is an element of A x ∉ A x is not an element of A |A| The number of elements in a finite set A A ⊆ B A is a subset of B ∅ The empty set A ∩ B The intersection of A and B A ∪ B The union of A and B B − A The complement of A relative to B Ac The complement of A relative to the universe A ⊕ B The symmetric difference of A and B A × B The cartesian product of A with B $\wp (A)$ The power set of A, the set of all subsets of A n! n factorial, the product of the first n positive integers $\binom{n}{k}$ n choose k, the number of k element subsets of a n set p ∧ q The conjunction, p and q p ∨ q The disjunction, p or q ¬p The negation of p, “not p" p → q The conditional proposition If p then q p ↔ q The biconditional proposition p if and only if q 1 The symbol for a tautology 0 The symbol for a contradiction r ⇐⇒ s r is logically equivalent to s r ⇒ s r implies s p | q The Sheffer Stroke of p and q Tp The truth set of p $(\exists n)\cup (p(n))$ The statement that p(n) is true for at least one value of n $(\forall n)\cup (p(n))$ The statement that p(n) is always true 0m × n The m by n zero matrix In The n × n identity matrix A−1 A inverse, the multiplicative inverse of A det A or |A| The determinant of A, 2 by 2 case a | b a divides b, or a divides evenly into b xsy x is related to y through the relation s rs The composition of relation r with relation s a ≡n b a is congruent to b modulo n a ≡ b (mod n) a is congruent to b modulo n c(a) the equivalence class of a under r r+ The transitive closure of r f : A → B A function, f, from A into B BA The set of all functions from A into B f(a) The image of a under f f(X) Range of function f : X → Y |A| = n A has cardinality n (g ◦ f)(x) = g(f (x)) The composition of g with f f ◦ f = f2 The "square" of a function i or iA The identity function (on a set A) f−1 The inverse of function f, read "f inverse" logba Logarithm, base b of a S ↑ S pop S ↓ S push S ∗ T Convolution of sequences S and T S ↑ p Multiple pop operation on S S ↓ p Multiple push operation on S Kn A complete undirected graph with n vertices deg(v), indeg(v), outdeg(v) degree, indegree, and outdegree of vertex v e(v) The eccentricity of a vertex d(G) The diameter of graph G r(G) The radius of graph G C (G) The center of graph G Qn The n-cube V (f) The value of flow f Pn A path graph of length n $\chi (G)$ The chromatic number of G Cn A cycle with n edges ∗ The generic symbol for a binary operation string1 + string2 The concatenation of string1 and string2 [G; ∗] A group with elements G and binary operation gcd(a, b) The greatest common divisor of a and b a +n b The mod n sum of a and b a ×n b The mod n product of a and b $\mathbb{Z}_n$ The Additive Group of Integer Modulo n $\mathbb{U}_n$ The Multiplicative Group of Integer Modulo n W ≤ V W is a subsystem of V ⟨a⟩ The cyclic subgroup generated by a ord(a) Order of a V1 × V2 × · · · × Vn The direct product of algebraic structures V1, V2, ..., Vn G1 ≅ G2 G1 is isomorphic to G2 dim(V) The dimension of vector space V 0 The least element in a poset 1 The greatest element in a poset Dn The set of divisors of integer n a ∨ b The join, or least upper bound of a and b a ∧ b The meet, or greatest lower bound of a and b [L; ∨, ∧] A lattice with domain having meet and join operations $\bar{a}$ The complement of lattice element a [B; ∨, ∧, ¯ ] A boolean algebra with operations join, meet, and complementation $M_{\delta_1\delta_2...\delta_k}$ The minterm generated by $x_1,x_2,...,x_k$ where $y_i=x_i$ if $\delta_i=1$ and $y_i=\bar{x}_i$ if $\delta_i=0$ A∗ The set of all strings over an alphabet A An The set of all strings of length n over an alphabet A $\lambda$ The empty string s1 + s2 The concatenation of strings s1 and s2 L(G) Language created by phrase structure grammar (S, X, Z, w, t) A finite-state machine with states S, input alphabet X, output alphabet Z, output function w, and next-state function t m(M) The machine of monoid M a ∗ H, H ∗ a The left and right cosets generated by a G/H The factor group G mod H SA The group of permutations of the set A Sn The group of permutations on a set with n elements An The Alternating Group Dn The nth dihedral group H ◁ G H is a normal subgroup of G kerθ The kernel of homomorphism θ [R; +, ·] A ring with domain R and operations + and · U (R) The set of units of a ring R D A generic integral domain deg f(x) The degree of polynomial f(x) R[x] The set of all polynomials in x over R R[[x]] The set of all powers series in R $\grave{x}, \acute{x}$ Pre and post values of a variable x M(A)i,j The i, j minor of A C(A)i,j The i, j cofactor of A det(A) or |A| The determinant of A This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.