This thesis gives a quick overview of applications in Compressed Sensing and focuses on the equivalence between non-convex ℓ⁰- and convex ℓ¹-minimization. In Compressed sensing, a solution to the equation Ax = b is wanted. The matrix A however has far more columns than rows, i.e. the linear system of equations is underdetermined. To still find a solution to this system, a sparsity assumption for the signal x is made. This leads to non-convex ℓ⁰- minimization of x subject to Ax = b. However, this problem is known to be NP-hard and therefore considered to be intractable. Researchers found out that solving the convex ℓ¹- minimization problem instead provides surprisingly good results. This thesis gives an overview of different conditions under which the two problems are equivalent and tests those different properties on small-scale examples. It will be seen that showing the equivalence for a given example is usually too hard. To this end, it is recommended to use the presented conditions to construct a matrix with good properties to ensure equivalence ofℓ¹- and ℓ⁰-minimization.