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Discrete Mathematics
42 flashcards
Combinatorics is a branch of mathematics that studies the counting and enumeration of discrete structures.
The fundamental counting principle states that if one procedure can be done in m ways and another can be done in n ways, then there are m * n ways of doing both procedures successively.
A permutation is an arrangement of elements in a definite order.
The number of permutations of n distinct objects is given by the formula: P(n) = n!
A combination is a selection of elements from a set, regardless of order.
The number of combinations of r elements from a set of n distinct objects is given by the formula: C(n, r) = n! / (r! * (n - r)!)
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
The two main types of graphs are undirected graphs and directed graphs.
A vertex (plural: vertices) is a fundamental unit of a graph, representing an object or entity.
An edge is a connection or line between two vertices in a graph, representing a relationship or interaction between the objects or entities.
A path in a graph is a sequence of vertices connected by edges, where no vertex is repeated.
A cycle in a graph is a closed path, where the starting and ending vertices are the same.
A tree is a connected, acyclic, undirected graph.
A spanning tree of a connected graph is a tree that includes all the vertices of the graph.
Logic is the study of the principles and criteria of valid inference and demonstration.
A proposition is a statement that can be either true or false, but not both.
The basic logical connectives are AND, OR, NOT, IMPLIES, and IFF (if and only if).
A truth table is a mathematical table used in logic to determine the truth value of a compound proposition based on the truth values of its components.
A tautology is a statement or compound proposition that is always true, regardless of the truth values of its components.
A contradiction is a statement or compound proposition that is always false, regardless of the truth values of its components.
Set theory is a branch of mathematical logic that studies sets, which are collections of objects.
The empty set, denoted by the symbol ∅ or {}, is a set that contains no elements.
The basic set operations are union, intersection, complement, and set difference.
A subset is a set whose elements are all members of another set.
The power set of a set is the set of all subsets of that set, including the empty set and the set itself.
An algorithm is a set of well-defined instructions or rules designed to solve a specific problem or accomplish a particular task.
The main properties of algorithms are input, output, definiteness, finiteness, and effectiveness.
An algorithm is a set of rules or instructions, while a program is an implementation of an algorithm in a specific programming language.
The time complexity of an algorithm is a measure of how the execution time scales with the input size.
The space complexity of an algorithm is a measure of how much memory or storage space is required by the algorithm to solve a problem.
The sorting problem is the task of arranging a set of elements in a specific order (e.g., ascending or descending) based on a comparison operation.
The searching problem is the task of finding a specific element or set of elements within a collection of data.
The divide-and-conquer algorithm design paradigm involves breaking a problem into smaller subproblems, solving those subproblems, and then combining the solutions to solve the original problem.
The greedy algorithm design paradigm involves making the locally optimal choice at each stage with the hope of finding a global optimum.
The dynamic programming algorithm design paradigm involves breaking a problem into smaller subproblems, solving the subproblems once, and storing the solutions to avoid recomputing them.
The backtracking algorithm design paradigm involves building candidates to the solutions and abandoning a candidate (backtracking) as soon as it determines that the candidate cannot possibly be completed to a valid solution.
A deterministic algorithm is an algorithm that produces the same output for a given input every time it is executed, while a non-deterministic algorithm may produce different outputs for the same input on different executions.
The halting problem is the decision problem of determining whether a given program will finish running (halt) or continue to run forever, given its input.
The P vs. NP problem is a fundamental question in computer science and computational complexity theory that asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
The traveling salesman problem is a classic optimization problem in computer science and operations research that involves finding the shortest possible route for a salesman to visit a set of cities and return to the starting point.
The knapsack problem is a problem in combinatorial optimization that involves packing the most valuable or important items into a container with limited capacity.
The graph coloring problem is a mathematical problem that involves assigning colors to the vertices of a graph such that no two adjacent vertices share the same color, while using the minimum number of colors.