Properties of Relations

Set Theory · Axiom Academy

LESSON Properties of Relations Reflexive, symmetric, antisymmetric, and transitive properties A relation R on set A is reflexive if every element is related to itself: A relation R on set A is symmetric if whenever x is related to y, then y is also related to x: A relation R on set A is antisymmetric if whenever both xRy and yRx hold, then x must equal y: This prevents "two-way" relationships between distinct elements. A relation R on set A is transitive if whenever x relates to y and y relates to z, then x relates to z: This creates a "chain" property: relations can be followed through intermediaries. Relations can have multiple properties simultaneously. Different combinations define important classes: Equivalence relation: Reflexive + Symmetric + Transitive Example: Equality, congruence mod n Partial order: Reflexive + Antisymmetric + Transitive Example: ≤ on numbers, ⊆ on sets Strict order: Antisymmetric + Transitive (not reflexive) Example: < on numbers

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