Need help with Discrete Math Question

I can't seem to figure out how to answer this question. I would really appreciate it if you can solve it for me. Here is the question: Let A ? [9]^(3) with |A| = 28. How small can the lower shadow of A be? And the upper shadow?

  • Can you define lower and upper shadow?

  • Lower shadow: is For A ⊂ X(r) , 1 <= r <= n, the shadow or lower shadow of A is the set system: ∂A = ∂ −A = {B ∈ X(r−1) : B ∪ {i} ∈ A for some i /∈ B} ⊂ X(r−1) E.g., if A = {123, 124, 234, 135}, then ∂A = {12, 13, 23, 14, 24, 34, 15, 35}. Upper shadow is the other way for example if elements are {12, 13, 23, 14, 24, 34, 15, 35} then the upper shadow is {123, 124, 234, 135}

The user who accepted to answer this question did not submit their answer before the deadline, and the question is now closed.