91Ó°ÊÓ

Prove. Any subset of a countable set is countable.

Prove. A set \(A\) is infinite if and only if there exists a bijection between \(A\) and a proper subset of itself.

Prove. The open interval \((a, b)\) is uncountable. [Hint: Find a suitable bijection from \((0,1)\) to \((a, b) . ]\)

Prove. The set \(Q^{+}\) of positive rational numbers is countable.

Prove. The set of irrational numbers is uncountable. (Hint: Prove by contradiction.)

Prove. A countable union of countable sets is countable.

Prove. The cartesian product of two countable sets is countable.

Prove. If \(\Sigma\) is a finite alphabet, then \(\Sigma^{*}\) is countable.

Prove each. If \(A\) and \(B\) are two invertible matrices of order \(n,\) then \((A B)^{-1}=B^{-1} A^{-1} .\)

Write an algorithm to compute the sum of the matrices \(A=\left(a_{i j}\right)_{m \times n}\) and \(B=\left(b_{i j}\right)_{m \times n}\).