Glossary
- Page ID
- 87143
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| (Eg. "Genetic, Hereditary, DNA ...") | (Eg. "Relating to genes or heredity") |  |
The infamous double helix | https://bio.libretexts.org/ | CC-BY-SA; Delmar Larsen |
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Word(s) |
Definition |
Image | Caption | Link | Source |
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| Antisymmetric | A relation R on a
set A is an antisymmetric relation provided that for all x,y∈Ax, if x R y and y R x, then x=y. |
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| Biconditional | P if and only if Q | ||||
| Bijection | A function f:A -> B such that f is both an injection and a surjection | ||||
| Bipartite graph | A graph for which it is possible to divide the vertices into two disjoint sets such that there are no edges between any two vertices in the same
set |
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| Boolean algebra | a lattice that contains a least element and a greatest element and that is both complemented and distributive. The notation [B;∨,∧,¯] is used to denote the boolean algebra with operations join, meet and complementation. | ||||
| Cardinality | The number of elements in a set | ||||
| Chromatic number | The minimum number of colors required in a proper vertex coloring of the graph | ||||
| Complete graph | A graph in which every pair of vertices is adjacent | ||||
| Complement of a set | Let U be the universal set. Ac ={x in U, x is not in A} | ||||
| Conditional Statement | If P then Q. P is the antecedent (hypothesis) and Q is the consequent (conclusion) | ||||
| Congruent | Two integers are congruent mod n (n a positive integer) if the integers leave the same remainder upon division by n | ||||
| Conjecture | a guess in mathematics | ||||
| Conjunction | P and Q | ||||
| Contrapositive | Of the conditional, if not Q then not P, logically equivalent to if P then Q | ||||
| Converse | Of the conditional, if Q then P | ||||
| Countably infinite | A set that can be put into one-to-one correspondence with the set of natural numbers | ||||
| Cyclic group | Group G is cyclic if there exists a∈G such that the cyclic subgroup generated by a, ⟨a⟩ equals all of G. That is, G={na|n∈Z} in which case a is called a generator of G. The reader should note that additive notation is used for G | ||||
| Digraph | A directed graph | ||||
| Direct Proof | Argument that is based on "If P then Q" and "P" implies Q | ||||
| Disjunction | P and Q | ||||
| Division algorithm |
Let m and d be integers with d>0. Then, there exists unique integers q and r with 0<=r < d such that m = dq+r |
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| Equivalence class | For each a∈A, the equivalence class of aa determined by ∼ is the subset of A, denoted by [a], consisting of all the elements of A that are equivalent to a |
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| Equivalence relation | A relation that is reflexive, symmetric and transitive | ||||
| Euclidean algorithm |
Let a and b be integers with a≠0 and b>0. Then gcd(a, b) is the only natural number d such that (a) d divides a and d divides b, and |
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| Euler circuit | An Euler path which starts and stops at the same vertex | ||||
| Euler path | A walk which uses each edge exactly once | ||||
| Existential operator | There exists | ||||
| Formal Language | If A is an alphabet, a formal language over A is a subset of A∗. | ||||
| Graph | an ordered pair G=(V,E) consisting of a nonempty
set V (called the vertices) and aset E (called the edges) of two-element subsets of V |
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| Greatest common divisor | The largest natural number that divides both a and b is called the greatest common divisor of a and b |
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| Hasse diagram | an illustration of a poset | ||||
| Homomorphism | Let [G;∗] and [G′;⋄]be groups. θ:G→G′ is a homomorphism if θ(x∗y)=θ(x)⋄θ(y) for all x,y∈G | ||||
| Indirect Proof | Argument based upon the contrapositive | ||||
| Injection | A function f:A->B is an injection, if for every a and b in the domain of f, f(a)=f(b) implies a = b. | ||||
| Intersection of two sets | Let U be the universal set. A intersect B = {x in U, x is in A and x is in B} | ||||
| Inverse | Of the conditional, if not P then not Q | ||||
| Isomorphism | between two graphs G1 and G2 is a bijection f:V1→V2 between the vertices of the graphs such that if {a,b} is an edge in G1 then {f(a),f(b)} is an edge in G2 |
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| Lattice | a poset (L,⪯)for which every pair of elements has a greatest lower bound and least upper bound | ||||
| Logical equivalence | Two expressions are logically equivalent provided that they have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions | ||||
| Monoid |
a set M together with a binary operation ∗∗ with the properties
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| Negation | not P | ||||
| Partial order |
Let ⪯⪯ be a relation on aset L.L. We say that ⪯⪯ is a partial ordering on LL if it isreflexive ,antisymmetric , andtransitive . That is:
The set together with therelation (L,⪯) is called a poset. |
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| Principle of Mathematical Induction |
A technique of proof whereby If T is a subset of N such that
Then T=N. |
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| Proof by Contradiction | Given "If P then Q" assume "P and not Q" to arrive at "P and not P" | ||||
| Reflexive | The relation R is reflexive on A provided that for each x∈A, x R x or, equivalently, (x,x)∈R |
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| Relation | Let A and B be sets. A relation R from the
set A to the set B is a subset of A×B |
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| Set | A well-defined collection of objects | ||||
| Statement | a declarative sentence that is either true or false but not both | ||||
| String | A string of length n, n⩾1 over alphabet A is a sequence of nn letters from A: a1a2…an. The set of all strings over A is A*. |
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| Subgraph | We say that G1=(V1,E1) is a subgraph of G2=(V2,E2) provided V1⊆V2 and E1⊆E2 | ||||
| Surjection | A function f:A ->B is a surjection if for every b in B there exists an a in A such that f(a)=b. | ||||
| Symmetric |
The relation R is symmetric provided that for every x,y∈A, if x R y, then y R x or, equivalently, for every x,y∈Ax, if (x,y)∈R, then (y,x)∈R |
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| Symmetric group | Let A be a nonempty set . Theset of all permutations on A with the operation of function composition is called thesymmetric group on A, denoted SA. |
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| Tree | A (connected) graph with no cycles. (A non-connected graph with no cycles is called a forest.) The vertices in a tree with degree 1 are called leaves | ||||
| The Well-Ordering Principle | for the natural numbers states that any nonempty
set of natural numbers must contain a least element equivalent to the Principle of Mathematical Induction |
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| Transitive | The relation R is transitive provided that for every x,y,z∈A, if x R y and y R z, then x R z or, equivalently, for every x,y,z∈A, if (x,y)∈R and (y,z)∈R, then (x,z)∈R | ||||
| Truth table | a summary of all the truth values of a statement | ||||
| Uncountably infinite | An infinite set for which there does not exist a one to one correspondence with the natural numbers | ||||
| Union of two sets | Let U be the universal set. A U B ={x in U, such that x is in A or x is in B} | ||||
| Universal operator | For all or for every | ||||
| Vertex coloring | An assignment of colors to each of the vertices of a graph. A vertex coloring is proper if adjacent vertices are always colored differently |