We define in this work a notion of Young differential inclusion dz t P F pz t qdx t , for an α-Hölder control x, with α ą 1{2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, γ-Hölder continuous set-valued map on the interval r0, 1s has a selection with finite p-variation, for p ą 1{γ. We also give a notion of solution to the rough differential inclusion dz t P F pz t qdt`Gpz t qdX t , for an α-Hölder rough path X with α P`1 3 , 1 2 ‰ , a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.