Binary relation in the context of "Reflexive relation"

In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is an operator that combines or modifies one or more logical variables or formulas, similarly to how arithmetic connectives like ${\displaystyle +}$ and ${\displaystyle -}$ combine or negate arithmetic expressions. For instance, in the syntax of propositional logic, the binary connective ${\displaystyle \lor }$ (meaning "or") can be used to join the two logical formulas ${\displaystyle P}$ and ${\displaystyle Q}$, producing the complex formula ${\displaystyle P\lor Q}$.

Unlike in algebra, there are many symbols in use for each logical connective. The table "Logical connectives" shows examples.

A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if:

where the notation aRb means that (a, b) ∈ R.

In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x = y and y = z then x = z.

In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset (or subclass) S ⊆ X has a minimal element with respect to R; that is, there exists an m ∈ S such that, for every s ∈ S, one does not have s R m. More formally, a relation is well-founded if:$${\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].}$$Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.

Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence x₀, x₁, x₂, ... of elements of X such that x_n+1 R x_n for every natural number n.

In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from existential quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain.

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

A function is bijective if it is invertible; that is, a function ${\displaystyle f:X\to Y}$ is bijective if and only if there is a function ${\displaystyle g:Y\to X,}$ the inverse of f, such that each of the two ways for composing the two functions produces an identity function: ${\displaystyle g(f(x))=x}$ for each ${\displaystyle x}$ in ${\displaystyle X}$ and ${\displaystyle f(g(y))=y}$ for each ${\displaystyle y}$ in ${\displaystyle Y.}$

Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".