Definition·D042
Parallel lines
Lines that are equal or never meet.
for lines
of an affine plane, with intersection and the empty set.
In words
Two lines are parallel when they are the same line or they share no point at all.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Counting a line as parallel to itself is the standard convention in incidence geometry: it makes parallelism reflexive, and the deep content, transitivity, becomes T14. As a set, the relation is
by Separation on the product, an honest relation on
. Playfair's axiom (AP2) of D041 can then be read: through every point not on
passes exactly one line parallel to
.
Used by
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