Final answer:
The question involves using properties of ceiling and floor functions in mathematical proofs and disprovals. A clear example is provided to disprove the incorrect generalization of the ceiling function for a product, but the proofs part cannot be completed due to incomplete information.
Step-by-step explanation:
The question involves proving two mathematical statements involving ceiling and floor functions and disproving another one. For the proofs, we would typically rely on the properties of the ceiling and floor functions along with algebraic manipulation. Unfortunately, due to the presence of typos or incomplete information in the question provided, a valid complete proof cannot be given. However, to address a correctable part of the question regarding the ceiling function:
1) The statement ⋄x∈R,⌈x-1⌉=⌈x⌉-1 is true because subtracting 1 from x before applying the ceiling function will yield the same result as applying the ceiling function to x first and then subtracting 1, assuming x is not an integer. If x is an integer, ⌈x⌉ = x and ⌈x-1⌉ = x-1, which also satisfies the statement.
2) For disproving the statement ⋄x,y∈R,⌈xy⌉=⌈x⌉⌊floor⌋(y⌋), consider examples where x or y are not integers. If x is a fraction and y is an integer, the product xy could result in a non-integer that when applying the ceiling function would round up to the next integer, which wouldn't be the same as multiplying the ceiling of x (which would be the rounded up integer of x) with the floor of y (which would be y itself if y is an integer).