discrete mathematics
Proposition: A statement that is either true or false. The truth value of a proposition may be true or false.
Atomic propositions: On the basis of propositions, they cannot be decomposed into simpler sentences.
Compound propositions: Consists of atomic propositions and connectives.
Atomic formula: A single propositional variable or propositional constant is called an atomic formula.
Rules for a well-formed formula ::
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A single atomic formula is a well-formed formula,
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If A is a well-formed formula, then not-A is also a well-formed formula,
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If A and B are well-formed formulas, then A conjuncts B, A disjuncts B, A implies B, and A is equivalent to B are all well-formed formulas.
A limited number of uses of these three rules is a valid formula.
Eternal truth formula: also known as tautology, a formula that is true for all truth value assignments and definite truth values is called eternal truth formula.
Perpetual false formula: also known as contradictory formula, a formula that assigns all truth values to false is false, then this formula is called permanent false formula.
Satisfiable formula: A formula with at least one truth assignment that makes the formula true is called a satisfiable formula.
Equivalence: For formulas A and B under any assignment, the truth values are the same, then A and B are said to be equivalent.
Paradigm: A standard that unifies well-formed formulas in various forms into a normative form.
Substitution rule: If A is a tautology, if any formula is used to replace a propositional variable in A, the new formula obtained is still a tautology.
Substitution rules: A1 is a sub-formula of A, if A1 and B1 are equivalent, then replace A1 with B1 to get formula B and formula A are equivalent.
Disjunctive normal form: disjunctive form of a finite number of phrases
Conjunctive Normal Form: Conjunctive Forms with a Limited Number of Clauses
Paradigm existence theorem: For any propositional formula, there are disjunctive normal forms and conjunctional normal forms equivalent to it.
Simple conjunction: A conjunction consisting only of propositional variables and their negations is called a simple conjunction.
Simple disjunction: A disjunction consisting only of propositional variables and their negations is called a simple disjunction.
Minimal term: For a simple conjunction with n propositional variables, each propositional variable and its negation do not exist at the same time, and only appear once, such a conjunction is called a minimal term.
Maximum term: For a simple disjunction with n propositional variables, each propositional variable and its negation do not exist at the same time, and only appear once, such a disjunction is called a maximum term.
Master disjunctive normal form: a disjunctive normal form composed of minimal terms.
Principal Conjunctive Normal Form: A conjunctional normal form composed of maximal terms.
Closed form: There are no freely appearing individual variables, and all individual variables are constrained to appear.
The nature of the set: disorder, dissimilarity, determinism (as long as the set is given, it is determined whether an element belongs to the set), arbitrariness (the elements in the set can also be a set).
Set relations: equality, containment, true containment.
Power set: The set of all subsets of a set.
Representation method of relationship: relationship matrix, relationship graph.
The relation matrix of the empty relation is an all-zero matrix, the relation matrix of the global relation is an all-one matrix, and the identity relation is an identity matrix.
Symmetric relations: Symmetric matrices, such as global relations, identity relations, and null relations.
Antisymmetric relationship: asymmetric matrix, any two elements that are symmetric about the main diagonal cannot be 1 at the same time.
Reflexive relationship: all 1s on the main diagonal, such as global relationship, identity relationship, less than or equal relationship, divisible relationship
Anti-reflexive relationship: the main diagonal is all 0, such as the less than relationship
The relationship matrices of the inverse relationship are transposed matrices.
Closure
A new relation R1 is obtained by adding as few elements as possible to the relation R, so that R is reflexive, and R1 is called a reflexive closure.
Add as few elements as possible to the relation R to get a new relation R1, which makes R symmetric, and R1 is called a symmetric closure.
Add as few elements as possible to the relation R to get a new relation R1, so that R is transitive, and R1 is called transitive closure.
Equivalence relation : A binary relation R on a non-empty set has reflexivity, symmetry, and transitivity at the same time, then R is an equivalence relation on a non-empty set.
Equivalence class: A set is divided by an equivalence relationship, each part of the division is an equivalence class, and each element equivalent to an element constitutes an equivalence class.
Properties: The equivalence classes of equivalent elements are the same, the equivalence classes of unequal elements are disjoint, and the union of equivalence classes of all elements is the original set.
Business set : A set with all equivalence classes as elements becomes the business set of this set about this equivalence relationship.
Partition : A family of subsets of non-empty sets of set A, different partitions are mutually disjoint, and the union of these partitions is set A.
Partial order relationship : The relationship R on a non-empty set is reflexive, antisymmetric, and transitive, so R is a partial order relationship on this set.
Covering relationship : A and B are comparable, BRA, and there is no C, so that BRC, CRA, then B can cover A.
Upper bound: greater than or equal to all elements in the subset
supremum: the smallest upper bound
Total order relationship : R is a partial order relationship on the set A. Any two elements x, y in A always have xRy or yRx, then R is said to be a total order relationship.
Well-ordered relationship : R is a partial order relationship on the set A. If any non-empty subset in A has the smallest element, then R is said to be a well-ordered relationship.
map
Surjective: codomain = value domain, that is, the mapping of A->B, each element in B is obtained by a certain element in A through function mapping.
Injection: A->B mapping, for all elements in A, B has a unique element corresponding to it.
Double shot: Both full shot and single shot.
Cardinality or cardinality: The number of elements in a set.
Finite set: A collection of finite elements.
Infinite set: A collection that is not a finite set.
Countably Infinite Sets: Aleph Zero Integers, Natural Numbers, Rational Numbers
Uncountable infinite sets: real numbers, irrational numbers
Algebraic system : A system composed of a non-empty set together with operations defined on the set is called an algebraic system. An algebraic system needs to satisfy that operations are closed on sets.
The law of idempotence: The element x in the set A, x is operated on itself, and the result is still x, then the operation is said to be idempotent.
Unity: left and right units are equal, left: e * x = x, right: x * e = x
Nulls: left and right zeros are equal, left: a * x = a, right: x * a = a
Inverse elements: left and right inverse elements are equal, left: a * b = e, right: b * a = e
Idempotent element: If a * a = a in the set A, then a is an idempotent element of this algebraic system.
Eliminable element: if a * x = a * y, then x = y, and a is called an algebraic system that can be eliminated.
Semigroups : Algebraic systems that satisfy associative laws.
Subsemigroup : The subset of the semigroup set is also closed for operations, then the algebraic system formed by the subset and operations is also a semigroup, and the latter is a subsemigroup of the former.
Commutative semigroup : A semigroup that satisfies the commutative law.
Containing a monoid : a semigroup with a unit, also known as a unique point.
Group : An algebraic system that satisfies closure, associativity, has a unitary element, and has an inverse for each element is a group, that is, a monooidal group that satisfies that each element has an inverse is a group.
The elements in the group all have inverse elements and no zero elements, satisfying the law of elimination.
A finite semigroup that satisfies the elimination law is a group.
Ordinary subgroups : There must be two subgroups in a group above the second order, one is a subgroup composed of unitary elements, and the other is the group itself. These two subgroups are collectively called ordinary subgroups, and the other subgroups are non-trivial subgroups.
Commutative group : A group that satisfies the commutative law is called a commutative group.
Cyclic group : There is a generator a, and each element in the group can be obtained by a finite number of operations of the generator. Such a group is called a cyclic group. Generators may not be unique.
Any cyclic group is a commutative group.
Lattices : In a partially ordered set, if there exists any a, b belonging to this set, {a, b} has supremum and infimum, then this partially ordered set is said to be a lattice, and the supremum of {a, b} It is called the union of a and b, and the infimum of {a, b} is called the intersection of a and b.
Properties: Closed, intersection and union operations all satisfy commutative law, associative law, and absorption law.
Distributive lattices : The lattices whose intersection and union operations satisfy the distributive law are called distributive lattices. The most typical ones that are not distributive lattices are diamond lattices and pentagonal lattices.
Bounded lattice : A lattice with an upper bound such that all elements are smaller than this upper bound and a lower bound such that all elements are smaller than this lower bound is called a bounded lattice.
The most typical ones are not bounded lattices, such as integers as sets, and less than lattices as operations.
Complementary element : For a bounded lattice < A, <= >, a is an element in the set A, if there is b belonging to A, so that a and b=1, a intersecting b = 0, then b is the complement of a.
There can be no supplementary element, or more than one.
Complementary grid : Every element in a bounded grid has a complement, so this bounded grid is a complementary grid.
The supplementary element in the allocation grid is unique.
Boolean : Complemented lattice, each element has a unique complement
Mixed graph : Some edges are directed and some edges are undirected, then this graph is called a mixed graph.
Base or base graph : In a directed graph, if each directed edge is changed to an undirected edge, the base or base graph of the directed graph is obtained.
Zero graph : n vertices, no edges
Trivial graph : 1 vertex, no edges
Multigraphs : graphs with parallel edges
Line Graphs : Non-Multiple Graphs are called Line Graphs
Simple graphs : graphs without parallel edges and self-loops
Isomorphism of graphs : two graphs are both directed or undirected graphs, and the point sets of the two graphs have bijections, and there are corresponding edges between corresponding points, and the direction of the directed edges is also the same, then the two graphs are called Graph isomorphism.
Subgraph : Delete some points or edges, or both points and edges from the original graph.
True subgraph : Same as subgraph, but it is not allowed to delete nothing.
Generate subgraph : Same as subgraph, only edge deletion is allowed, and point deletion is not allowed.
Self-complementary graph : A graph is self-complementary if it is isomorphic to its complement.
Simple paths : paths with different edges; simple circuits are circuits with different edges
Basic path : a path with different nodes, and a basic circuit is a circuit with different nodes.
The basic way must be the simple way.
Point-cut set : a connected graph, if you delete all the elements in the point-cut set, it is no longer a connected graph, and if you delete any proper subset in the point-cut set, it is still a connected graph.
Cut point : A node constitutes a point cut set.
Node connectivity = the number of elements in the point cut set
Edge-cut set : A connected graph, if you delete all the elements in the edge-cut set, it is no longer a connected graph, but if you delete any proper subset of the edge-cut set, it is still a connected graph.
Edge cut : An edge constitutes an edge cut set.
Edge connectivity = the number of elements in the edge cut set
One-way connectivity : between any two nodes, one node can reach another node.
Weak connectivity : the directed graph is not unidirectionally connected, but the underlying graph is connected
Strongly connected branch : The subgraph G of the directed graph is strongly connected, but if some points and edges are added, it is no longer strongly connected, so G is a strongly connected branch.
Euler Path : A path that passes each edge exactly once can be found.
Euler circuit : It is possible to walk out a circuit that passes each edge only once.
Euler diagram : A graph with an Euler circuit.
Judgment of Euler path : There are only two or no odd-degree vertices in the graph.
Judgment of Euler circuit : There are no odd-degree vertices in the graph.
Determination of directed Euler circuit : the out-degree and in-degree of each vertex are the same.
Hamiltonian path : A path that each vertex passes through and only once.
Hamiltonian circuit : A circuit in which each vertex is traversed exactly once.
Hamiltonian Graph : A graph with a Hamiltonian cycle.
The judgment of the Hamiltonian graph : the sum of the degrees of any two vertices is greater than or equal to n-1, then the graph is a Hamiltonian graph.
Check whether you have reviewed well, look at the following keywords, and see if you can say the content. During the retest, the teacher often says a keyword, and then you expand according to the keyword, which can test both the breadth and the depth.
Always
true form Always false form
Satisfiable form Equivalent
form Paradigm
form Substitution rule Substitution rule Disjunctive normal form Conjunctive normal form Paradigm existence theorem Simple conjunction Simple disjunction Minimal term Maximal term Principal disjunctive normal form Principal conjunctive normal form Closed form Properties of sets, relations of sets, power sets , expressing methods of relations, equivalence classes , properties of equivalence classes, quotient sets , division , partial order relations, covering relations , upper bounds, lower bounds, infimum bounds, supremum relations, total order relations, well order relations, surjective, injective , bijective Cardinality Finite set Infinite set Algebraic system Idempotent law Unary zero-element inverse element Idempotent element Eliminable element Semigroup Subsemigroup Commutative semigroup containing monoid Group Ordinary subgroup commutative group Cyclic group Lattice distributive lattice Complementary Boolean Lattice Bounded Lattice Mixed Graph Base Graph Zero Graph Ordinary Graph Multiple Graph Line Graph Simple Graph Isomorphism of Graphs
Subgraph true subgraph generates subgraph self-complementary graph
Simple road basic road
Point cut set cut point point connectivity edge cut set cut edge connectivity
one-way connected weak connected strong connected
Euler graph Euler circuit Euler graph decision
Hamiltonian graph The Judgment of Hamiltonian Graph in Hamiltonian Circuit