If X,Y, and Z are sets and |X|=|Y| and |Y|=|Z|, show that |X|=|Z|. Note that we are not assuming that the given sets are finite.

Let assume sets X, Y, Z are not finite.

Since if we take X, Y, and Z are finite then result is trivial.

Now, we have |X| = |Y| and |Y| = |Z|.

Then we can define a bijection from X to Y says f

f : X → Y is bijection

Also there is a bijection g from Y onto Z

g : Y → Z is bijection

The composition of two bijective functions is bijective.

This shows that g○f is bijection from X to Z.

Since rang of f and domain of g is the same set, so the composition mapping is well defined.

Let h = g○f

Thus h is a bijection from set X onto set Z.

h : X → Z is bijection

|X| = |Z|