Question

Show that if a,y,z are rational numbers such that x < y and y < z, then x < z.

Answer 1

Order properties of Q(set of rational numbers)
On the set Q, a linear order relation < is defined by "a < b if a€Q,b€Q and a is less than b" and it satisfies the following conditions.
O1. If a, b €Q then exactly one of the following statements holds-
a<b, or a = b, or b<a; (law of trichotomy)
O2. a < b and b<c=> a< c for a, b, c€ Q; (transitivity)
O3. a <b=> a+c<b+c for a, b, c € Q;
O4. a < b and 0 <c=>ac < bc for a, b, c € Q.

Therefore this comes from transitivity.