In this Bachelor’s thesis the following question is answered: Does the inequality posed in the article Klyachko et al [2008] cover the real part of the Bloch surface of a 3D quantum system when used as in Kochen and Specker [1967]? The Klyachko inequality relies on using five measurements to show contextuality of a subset of states on the real part of the Bloch surface. These can now be used in several configurations as present in the Kochen-Specker contextuality proof, by simply rotating the measurements. We show here that these new inequalities will have subsets of violation that eventually cover the entire real part of the Bloch surface. This can be extended to show that all states of a spin 1 system are non-contextual, so that we have recovered a state-independent contextuality proof by using the Klyachko inequality several times. In the final part, an interpretation of this is given and also some recommendations for further research that should be done in the field.