22 Basic shapes of geometry Ch. 2
or
l∩m is a singleton,or
l=m and in this last case l∩m=l=m.Moreover the planeΠis not a line, as each line is a proper subset ofΠ.
If three or more points lie on one line we say that these points arecollinear.If
one point lies on three or more lines we say that these lines areconcurrent.
2.1.2 Naturalorderonaline.........................
COMMENT. The two intuitive senses of motion along a line give us the original
examples of linear (total) orders, and we refer to these as the two natural orders on
that line. On a diagram the sense of a double arrow gives one natural order onl, while
the opposite sense would yield the other natural order onl. We now take natural order
as a primitive term, and go on to define segments and half-lines in terms of this and
our existing terms.
A
B
Figure 2.1. A lineAB.
The arrows indicate that the line
is to be continued unendingly.A
B
Figure 2.2. The double arrow indicates
a sense along the lineAB.Primitive Term. On each linel∈Λthere is a binary relation≤l, which we refer
to as anatural orderonl. We readA≤lBas ‘Aprecedes or coincides withBonl’.
AXIOM A 2 .Each natural order≤lhas the properties:-
(i)A≤lA for all points A∈l;
(ii) if A≤lB and B≤lCthen A≤lC;
(iii)if A≤lB and B≤lA, then A=B;(iv) for any points A,B∈l, either A≤lBor B≤lA.|COMMENT. We refer to (i), (ii), (iii) in A 2 as the reflexive, transitive and antisym-
metric properties, respectively, of a binary relation; property (iii) can be reworded as,
ifA≤lBandA=BthenB≤lA. A binary relation with these three properties is
commonly called a partial order. A binary relation with all four properties (i), (ii),
(iii) and (iv) in A 2 is commonly called a linear order or a total order.