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preceq

Denotes a binary relation meaning 'precedes or equals' in mathematical ordering and set theory.

Overview

Commonly used in mathematical proofs and formal logic to express partial ordering relationships between elements or sets.

  • Essential in order theory for describing relationships between comparable elements
  • Frequently appears in abstract algebra and lattice theory
  • Used alongside similar relations like \prec and \succeq to establish formal orderings
  • Particularly useful in describing subset relationships with additional ordering constraints

Examples

Expressing a partial ordering relation between sets.

A⪯B  ⟺  A⊆BA \preceq B \iff A \subseteq B 
A \preceq B \iff A \subseteq B

Comparing elements in a partially ordered set.

x⪯y⪯zx \preceq y \preceq z 
x \preceq y \preceq z

Defining an inequality with a less than or equal to relation in ordered spaces.

f(x)⪯g(x) for all x∈Xf(x) \preceq g(x) \text{ for all } x \in X 
f(x) \preceq g(x) \text{ for all } x \in X