If ∀A ∈ PX , ∀B ∈ PY , |A| < |B| Then ∃A ∈ PX , ∃B ∈ PY , A ⊂ B

Question

X and Y are non-empty sets , X c Y (subset ) and Px is a partition of X and Py is a partition of Y .

How can I prove that this statement is true or false :

If ∀A ∈ PX, ∀B ∈ PY ,|A| < |B|

Then ∃A ∈ PX, ∃B ∈ PY , A ⊂ B

Thank you .

Ps: what I understood is that since XcY then A that belongs to X belongs to Y as well and since the cardinality of A is smaller than B then A is likely to be a subset of B . My guess is this statement is true but I have no idea how to prove it , which method to use ? I don’t know how to start my proof .

Answer 1

X = { 1,2,3,4,5,6 }.
Y = { 1,2,3,4,5,6,7 }.
PX = { {1,4}, {2,5}, {3,6} }.
PY = { {1,2,3}, {4,5,6,7} }.
What does that example show?