Finite Element problem - Understanding Convergence, Completeness and Compatibility?

I am reading this book Introduction to finite element methods by Niels Ottosen and Hans Petersson. In chapter 7 it brings up general requirements for FEM, such as convergence, completeness and compatibility. It states some definitions such as:

1. Its complete when the approximation is represented by an arbitrary constant temperature gradient and constant temperature.

2. Its compatibility when the approximation of the temperature over element boundaries is continuous.

3. Its convergence when completeness and compatibility is fulfilled.

Okay so far, but my problem is how to actually apply these definitions to an example?

What has pascals triangle with this?

Could someone help me understand what they say and mean with an example?

Say for a 6-node element, triangle with node at ends and one in the middle.

Approximation T = a1 + a2x +a3y + a4xy + a5x^2 + a6y^2?