proper subset in a sentence

Examples

1. Similarly, the idea that ? is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.
2. There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but in a distributive lattice the two types of elements coincide.
3. Using the proper subset of hardware description language, a program called a synthesizer, or clock edges as the way to time the circuit, ignoring any timing constructs.
4. The LL ( 1 ) languages are a proper subset of the LR ( 1 ) languages which in turn are a proper subset of all context-free languages.
5. The LL ( 1 ) languages are a proper subset of the LR ( 1 ) languages which in turn are a proper subset of all context-free languages.
6. The same is true for agent Y . Since S is a proper subset of the agents, the induction assumption can be invoked and the theorem is proved.
7. A constant function on, if has more than 1 element, has an image that is a proper subset of its codomain, is not bijective ( and non invertible ).
8. An unsatisfiable core is called a " minimal unsatisfiable core ", if every proper subset ( allowing removal of any arbitrary clause or clauses ) of it is satisfiable.
9. Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it.
10. The set of theorems of a deviant logic can differ in any possible way from classical logic's set of theorems : as a proper subset, superset, or fully exclusive set.