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In other words. $R$ is antisymmetric iff\n\n$(aRb ∧ bRa) → a = b$\n\nor, equivalently, $a ≠ b → (¬aRb ∨ ¬bRa)$\n\nAntisymmetry isn't quite the [set_theory_compliment compliment] of [symmetric_relation Symmetry]. Due to the fact that $aRa$ is allowed in an antisymmetric relation, the equivalence relation, $\\{(0,0), (1,1), (2,2)...\\}$ is both symmetric and antisymmetric.\n\nExamples of antisymmetric relations also include the successor relation, $\\{(0,1), (1,2), (2,3), (3,4)...\\}$, or this relation linking numbers to their prime factors $\\{...(9,3),(10,5),(10,2),(14,7),(14,2)...)\\}$\n\n', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: [ '0', '0', '0', '0', '0', '0', '0', '0', '0', '0' ], muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: { Summary: 'An antisymmetric relation is a relation where no two distinct elements are related in both directions. In other words. $R$ is antisymmetric iff' }, creatorIds: [ 'MYass', 'EricRogstad' ], childIds: [], parentIds: [ 'relation_mathematics' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [ { id: '5656', parentId: 'antisymmetric_relation', childId: 'antisymmetric_relation', type: 'subject', creatorId: 'MYass', createdAt: '2016-07-27 02:09:01', level: '3', isStrong: 'true', everPublished: 'true' } ], learnMore: [], requirements: [ { id: '5655', parentId: 'relation_mathematics', childId: 'antisymmetric_relation', type: 'requirement', creatorId: 'MYass', createdAt: '2016-07-27 02:08:37', level: '2', isStrong: 'false', everPublished: 'true' } ], subjects: [ { id: '5656', parentId: 'antisymmetric_relation', childId: 'antisymmetric_relation', type: 'subject', creatorId: 'MYass', createdAt: '2016-07-27 02:09:01', level: '3', isStrong: 'true', everPublished: 'true' } ], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, 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