Answer to Question #148123 in Discrete Mathematics for Promise Omiponle

Question

Let Bn={(x, y)|0≤x≤n and 0≤y≤n}, where n is a nonnegative integer. Find U(n=0 to infinity)Bn and ∩(n=0 to infinity)Bn.

Answer 1

1) U(n=0 to infinity)Bn = {(x, y)|0≤x and 0≤y} = B.

Indeed, for all n Bn is a subset of B, therefore,

(1.1) U(n=0 to infinity)Bn = {(x, y)|0≤x and 0≤y} is a subset of B.

Let (x,y) be any element of B, then there exists n such that 0≤max{x,y}≤n. This means that (x,y) belongs to Bn. That is, any element of B belongs to some Bn, and

(1.2) B is a subset of U(n=0 to infinity)Bn.

Joining together (1.1) and (1.2), we conclude that U(n=0 to infinity)Bn = {(x, y)|0≤x and 0≤y} = B

2) ∩(n=0 to infinity)Bn = {(0,0)} = B₀

Indeed, for all n B₀ is a subset of Bn, therefore,

(2.1) ∩(n=0 to infinity)Bn includes B₀.

(2.2) ∩(n=0 to infinity)Bn is a subset of Bn for all n, particularly, it is a subset of B₀

Joining together (2.1) and (2.2), we conclude that ∩(n=0 to infinity)Bn = B₀ = {(0,0)}